1.10: The Verb and μι-Verbs εἰμί, δίδωμι, τίθημι
Module 9
© 2021 Philip S. Peek, CC BY 4.0 https://doi.org/ 10.11647/OBP.0264. 10
The Verb
In Greek and in English, verbs have the same definition and functions. Verbs are words that represent actions ( throw ) and states of being ( be or exist ). They differ in the same fundamental way that Greek nouns differ from their English counterparts: they use endings to create meaning in a way that English does not. The Greek verb ( ῥῆμα ) in its finite form has an ending that indicates what person and number the subject is. The Greek infinitive has an ending that indicates that it is unmarked for person and number .
In this module you will see how the Greek verb functions with three μι -verbs: εἰμί , δί δωμι , and τίθημι .
Finite Verbs
The finite verbs I run or we run have a person I or we and a number, singular I or plural we . Note that the combination of a verb with its subject forms a clause .
Infinitives
Infinitives are not marked for person or number. In English the infinitive is preceded by to . To run , to go , to walk , to carry , to mark , and to sing are all infinitives and are all unmarked for person and for number.
Intransitive and Transitive Verbs
Verbs are either intransitive or transitive . Intransitive verbs do not take a direct object . εἰμί I am is an intransitive verb . Transitive verbs take a direct object. ὁράω I see is a transitive verb. It takes a direct object. In the sentence, ὁρ άω αὐτόν I see him , αὐτόν is the direct object of the transitive verb ὁρά ω .
Some verbs can be intransitive or transitive depending upon their use in context. For example the English verb walk is intransitive in the sentence, I walk to the store . And it is transitive in the sentence I walk my dog to the store . Dog functions as the direct object of walk . Consider the verb smell . In the sentence, the flower smells good , the verb smell is intransitive and describes a quality of the flower’s aromatic existence. In the sentence I smell the sauteeing onions and garlic , the verb smell is transitive and represents an action that I perform. Onions and garlic function as the direct objects of smell . Module 30 has practice translating exercises with the μι -verb ἵστημι , which can be both intransitive and transitive.
Linking Verbs
Linking verbs connect two items in a sentence. The verb to be is the most common linking verb. In the sentence she is a doctor , the verb is connects the subject she and the predicate nominative doctor . In the sentence he is hungry , the verb is connects the subject he and the predicate adjective hungry . Some common linking verbs in English are these: appear, become, feel, grow, look, remain, seem, smell, sound, stay, taste, and turn. Some common linking verbs in Greek are these: γίγνομαι be, be born ; δοκέω seem ; εἰμ ί be ; κεῖμαι lie, remain ; and φαίνω seem . The two items connected by the linking verb are in the same case, which in Greek is typically nominative. In the examples below, the linking verbs are bolded and the words in the same case are underlined.
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ἰατρὸς γίγνεται ὁ Ἱπποκράτης . |
Hippokrates is a doctor . |
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τὸ δὲ δοκέει ἄριστον . |
It seems best . |
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ὄνομά μοί ἐστιν Φίλιππος . |
My name is Philip . |
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ἥσυχος κεῖμαι . |
I remain quiet . |
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μεγάλα τὰ τόξα φαίνεται . |
The bows seem large . |
Verb Tense
English and Greek have a variety of verb tenses which express time. Tense refers to the times of the past, present , and future . Each tense indicates when the action of the verb occurs. The present tense indicates action that is currently happening. The future tense refers to an event that will happen at some later point. The past tense refers to actions that have already occurred.
Tense-Aspect
Time (tense) refers to the past, present, and future. Aspect refers to whether an action is incomplete (imperfective), completed (perfective), or in a state (stative). The present tense stem, which is used to form the present and imperfect tenses, communicates an incomplete action. The beginnings and ends of the action are undefined. Thus the tense-aspect for the present is present ongoing or repeated, and the tense-aspect for the imperfect is past ongoing or repeated (imperfective). The aorist tense stem communicates a completed action as a single whole. The boundaries of the action’s beginning and end define the action as completed (perfective). The tense-aspect of the perfect and future perfect (not covered in this text due to its rarity) have a completed state that results from a previous completed action or have a completed state (stative) whose effects are still relevant. The future tense stems have a temporal value but no aspect, showing no distinction between the imperfective (incomplete) and perfective (completed) aspect ( CGCG 33.4–6).
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Incomplete Aspect. In the sentences, she picks flowers, and she was picking flowers, the aspect is called incomplete because the beginning and end of the action are undefined. Completed Aspect. In the sentence, she picked flowers, the aspect of picked is completed because the boundaries of the beginning and end of the action are defined. Stative Aspect. In the sentence, he is amazed, the aspect of is amazed is stative because the action of the verb results in the subject’s being in a certain state, here amazement. |
In sum the aspects are three: incomplete, completed, or stative. In English, aspect depends upon verb tense and context. In Greek, verb tense is a clearer determinant of aspect, though context still matters.
Verb Mood
English has these moods: the conditional, imperative, indicative, infinitive , and subjunctive .
The conditional mood uses the helping verbs could, might, should, would . For the conditional mood, the proposition (could, might, should, would) depends on a condition (if). In this sentence,
If I show you my weak side, would you take me home?,
upon the successful conclusion of the conditional if , the proposition may be fulfilled.
The imperative mood gives someone a command,
So kick back, dig, while we do it.
Here the mood of the verb kick gives a command that may or may not be fulfilled.
The indicative is the mood of fact. This mood makes declarations that are presented as true. In this sentence,
Generals are gathered in their masses,
we are to believe that the generals have crowded together.
The infinitive is unmarked for person and number. Ι t is classified as a verbal noun and is best understood by thinking of its function as completing or enhancing the meaning of adjectives, clauses, nouns, and verbs. This is why the infinitive is referred to as a complement. Sometimes classified as a mood, the infinitive is potential in meaning, ἐ ν δυνάμει , because its action may or may not be realized. In this phrase,
standing in line to see the show tonight,
there is no guarantee that the person standing in line will get to see the show.
The subjunctive mood communicates ideas that are contrafactual, hypothetical, or uncertain.
I wish you were here.
I ask that he be still a moment and this mystery explore.
Oi, where be the bar, John?
Be, explore, and were are all subjunctives. In the first sentence the were is contrafactual, indicating that the subject I is in fact not here . Contrast this contrafactual sentence with the factual one: we were two lost souls . The two weres are identical in form but not in function. Context enables us to understand the difference in meaning between them. Be and explore both refer to events in the future that may or may not be fulfilled. In the third sentence the be expresses uncertainty as to where the bar be.
Greek also has five moods: imperative (command); indicative (factual statements usually); infinitive (potential); optative (factual or hypothetical); and subjunctive (hypothetical). English often uses helping verbs such as may , might , should to express hypothetical or uncertain situations that Greek expresses through a verb’s mood.
Verb Voice
English has two voices: active , (she throws) and passive , (she is thrown). Greek has three voices: active , middle , and passive .
Active Voice . In the active voice the subjects perform the verb’s action. In the sentence
she throws,
she performs the act of throwing and so throws is active voice.
Passive Voice . In passive voice the action of the verb is performed upon the subject. In the sentence,
she is thrown,
the act of throwing is performed upon the subject she . The verb is thrown is passive voice.
Middle Voice . Middle voice expresses actions the subjects perform upon themselves or actions that the subjects are personally interested in. Greek uses the middle voice to translate the English sentence,
she washes herself,
λούεται .
Greek also uses the middle voice to translate the sentence,
she has her daughters educated ,
παιδεύεται τὰς θυγάτερας .
In sum the voices are three. The Greek active and passive voices function just as they do in English. English does not have a middle voice. The Greek middle voice, covered in more detail in Module 19 , often has a reflexive meaning: I see myself ; I wash my hands. Typically you will translate it via the English active voice.
Principal Parts
English and Greek verbs have principal parts . In English the principal parts are these: go, went, gone, going . Speakers of English use these principal parts to compose the variety of tenses, moods, and voices the English verb enjoys.
Greek verbs have these six principal parts:
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I. Form: |
ἄγω |
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English Equivalent: |
I lead, I am leading, I do lead |
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Identification: |
first person singular, present indicative active |
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II. Form: |
ἄξω |
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English Equivalent: |
I shall lead, I shall be leading |
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Identification: |
first person singular, future indicative active |
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III. Form: |
ἤγαγον |
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Translation: |
I led |
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English Equivalent: |
first person singular, aorist indicative active |
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IV. Form: |
ἦχα |
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English Equivalent: |
I have led |
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Identification: |
first person singular, perfect indicative active |
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V. Form: |
ἦγμαι |
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English Equivalent: |
I have led or I have been led |
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Identification: |
first person singular, perfect indicative middle or passive |
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VI. Form: |
ἤχθην |
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English Equivalent: |
I was led |
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Identification: |
first person singular, aorist indicative passive |
From these six principal parts are formed all the various tenses, moods, and voices the Greek verb enjoys.
Practice with Verbs in English. Read (or sing) this song by Little Richard and give as much information as you can about each verb. Check your answers with those in the Answer Key .
[Intro]
La, la, la da da da da da
[Verse 1]
I hope that I should live to see
When every man should know he’s free
Prayers and suffrage from the past
All make our heart to be this sad
[Chorus]
I got my duty rock and roll
Now everybody, everybody, everybody’s got to be free, y’all
[Refrain]
La, la, la da da da da da
[Verse 2]
Let’s get rid of that old man, hey hey hey hey
And bring our government up to date
It may seem very hard to do
Just open your mind, let love come through
[Chorus]
You hear me calling, hear my plea
Everybody, everybody, everybody’s gotta be free
Oh yes!
[Refrain]
La, la, la da da da da da
[Verse 3]
We did our thing and we paid our dues
And let’s get rid of these freedom blues
It may seem very hard to do
Just open your mind, let love come through
[Chorus]
I got my duty rock and roll
Now everybody, everybody, everybody’s got to be free, y’all
[Outro]
La, la, la da da da da
The purpose of this exercise is to have you think actively about how the English verb conveys meaning. In doing this exercise reflect on how much you intuitively know about your native language. Also look to transfer this knowledge to your understanding of the Greek verb.
The μι- Verb εἰμί
εἰμί frequently occurs. In all languages the verb to be is used so frequently that its forms resist change and they tend to be highly irregular. The finite forms of verbs indicate person and number. Because the forms of εἰμ ί are unique, it is difficult at first to discern the person and number of each form. When you learn δίδωμι , τίθημι , and the omega- or ω -verb , you will understand clearly how verb endings indicate person and number.
Memorize the below forms of εἰμί . Note that each form of the verb contains its own subject. The Greek language does have equivalents for the English pronouns I, you, he, she, it; we, you, they . You will learn them in detail in Module 18 . Greek often does not state them separately from the verb. Contrast εἰμί I am with ἐγὼ εἰμί I am. εἰμί without the pronoun ἐγώ occurs more frequently. When the ancient Greek pronoun is present, it is often there for emphasis or for contrast or for sound or for some other aesthetic or practical reason.
Memorize εἰμί so that you are able to recite and write the forms, going from Greek to English and vice versa. Remember that present refers to time now; that the indicative is used to state facts; and that in active voice the subject performs the verb’s action or state of being.
Present Indicative Active of ε ἰμί
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Verb Form |
Translation |
Person and Number |
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εἰμί |
I am |
1 st person singular |
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εἶ εἶς |
you are |
2 nd person singular |
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ἐστί ( ν ) |
he, she, it is; there is |
3 rd person singular |
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ἐσμ έν |
we are |
1 st person plural |
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ἐστέ |
you are |
2 nd person plural |
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εἰσί ( ν ) |
they are; there are |
3 rd person plural |
- The letter nu in the third person singular and plural is a nu-movable. It may be added to the ending when the following word begins with a vowel or at the end of clauses or verses. Otherwise it is left off.
Present Infinitive Active of εἰμί
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εἶναι |
to be |
unmarked |
- All forms but εἶ and εἶναι are enclitic . Enclitics are pronounced closely with the word that precedes them. Some common ones are the adverbs γε , ποθέν , ποι , ποτέ , που , πως , and τοι ; the conjunction τε ; the pronouns με , μοι , μου , σε , σοι σου , τι , and τις ; and the verbs εἰμί , φημί . Enclitics sometimes have an accent and sometimes do not. They can also affect the accent of the word that precedes them. How they do is covered in Part II of the 21st-Century series.
Practice Translating εἰμί . Translate the sentences below, which have been adapted from Lucian’s The Ass , ὁ Ὄνος . Remember the meanings and functions of the cases presented in Module 7 . Nominative case endings are bolded and genitive endings are italicized. Also in these sentences, the definite article, which will be learned systematically in Modules 11 and 12, is present in these forms: τ ῆς , τό , τῶν . Translate each one as the . Check your understanding with the translations in the Answer Key . Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
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Case |
Ending |
Function |
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Nominative |
- α , - η , - ης , - ο , - οι , - ον , - ος , - υ |
subject or predicate of the verb |
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Genitive |
- ας , - εως , - ης , - ου , - ων |
dependence; object of preposition; translated with adverb and adjective |
- ἐκ Ὑπάτ ης πόλ εως τ ῆς Θετταλί ας εἰμί .
- ἐκ τ ῶν ἄνω εἶ .
- πλησίον ἤδη τ ῆς πόλ εώς ἐσμεν .
- ἐγὼ μὲν ἔρομαι εἰ ἔνδον ἐστέ .
- τ ὸ δ ὲ οἰκίδι ον οὕτως μικρ όν ἐστι .
- ἐγὼ Ἄβροι ά εἰμι· ποῖ δέ καὶ κατά γῃ ;
- σὺ δὲ καὶ νέ η εἶ καὶ καλ ή ;
- τ ὸ χωρί ον ἐστί τ ῆς ὁδ οῦ τραχ ύ .
- σὺ δὲ νέ ος εἶ καὶ εὔτον ος παλαιστ ής ;
- οἱ δὲ οὐκ ἄπειρ οι τ ῆς τέχν ης εἰ σίν .
Adverbs and Verbs
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ἄνω up |
κατάγῃ you stay, you lodge |
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* εἰμί be, be possible |
πλησίον near + gen. |
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ἔνδον within |
* ποῖ where |
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* ἔρομαι I ask |
Adjectives, Nouns, Pronouns
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Νοminative Case |
Genitive Case |
English Equivalent |
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Ἄβροια |
Ἀβροίας |
Abroia (woman’s name) |
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ἄπειροι |
ἀπείρων |
inexperienced in + gen. |
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* ἐγώ |
ἐμοῦ |
I, me, mine |
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εὔτονος |
εὐτόνου |
fit, well-strung, vigorous |
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Θετταλία |
Θετταλί ας |
Thessaly |
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* καλή |
καλῆς |
beautiful, noble, good |
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* μικρόν |
μικρο ῦ |
small, little, short |
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* νέη |
νέης |
new, fresh, young |
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ὁδό ς |
ὁδοῦ |
road, path; journey |
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* οἱ δέ |
τῶν δ έ |
they, them, theirs |
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οἰκίδιον |
οἰκιδίου |
abode, house |
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παλαιστής |
παλαιστοῦ |
wrestler |
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* πόλις |
πόλεως |
city |
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τέχνη |
τέχνης |
skill, art |
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τραχύ |
τραχέος |
rough |
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τὰ ἄνω |
τῶν ἄνω |
the north |
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Ὕ πατα |
Ὑπάτης |
Hypata |
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χωρίον |
χωρίου |
place, area |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
The μι- Verbs δίδωμι and τίθημι
Though not as numerous as omega- or ω -verbs , μι -verbs occur frequently. μι -verbs conjugate differently from ω -verbs in some tenses, typically the present, imperfect, aorist, and sometimes the perfect. In conjugating these verb forms, learn which stem to combine with which ending. The stem is the base of the word to which the ending is joined. Once you have the correct stem and ending, combine them to create the correct form. You may memorize the stems and endings or you may learn how to combine stems and endings so that you can recognize the forms as you encounter them. In the readings that occur in the textbook, use your knowledge of δίδωμι give and τίθημι put, place to identify the present indicative active forms of other μι -verbs, including ἀπόλλυμι kill, lose; die ; δείκνυμι show, display ; ἵημι release, hurl, send ; (mid.) hasten and ἵστημι place, stand, make stand .
Present Tense Stems
Use the long vowel grade stem for the singular (I, you, he, she, it) and the short vowel grade for the plural (we, you, they). Note that the stems differ even within the same tense, mood, and voice.
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long vowel grade stem |
short vowel grade stem |
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(use for the singular) |
(use for the plural) |
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διδω - |
διδο - |
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τιθη - |
τιθε - |
Present Tense Endings
Add to the correct tense stem.
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Primary Active (use for the present tense) |
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S |
Pl |
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1 st |
- μι |
- μεν |
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2 nd |
- ς |
- τε |
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3 rd |
- σι ( ν ) |
- ᾱσι ( ν ) |
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Present Infinitive Active |
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- ναι |
Present Indicative Active of δίδωμι |
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Stem Vowel Grade |
Verb Form |
Eng. Equivalent |
Person and Number |
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long |
δίδωμι |
I give |
1 st person singular |
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long |
δίδως |
you give |
2 nd person singular |
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long |
δί δωσι ( ν ) |
he, she, it gives |
3 rd person singular |
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short |
δίδομεν |
we give |
1 st person plural |
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short |
δίδοτε |
you give |
2 nd person plural |
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short |
διδόᾱσι ( ν ) |
they give |
3 rd person plural |
Present Infinitive Active of δίδωμι |
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short |
διδόναι |
to give |
unmarked |
Present Indicative Active of τίθημι . |
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Stem Vowel Grade |
Verb Form |
Eng. Equivalent |
Person and Number |
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long |
τίθημι |
I put |
1 st person singular |
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long |
τίθης |
you put |
2 nd person singular |
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long |
τίθησι ( ν ) |
he, she, it puts |
3 rd person singular |
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short |
τίθεμεν |
we put |
1 st person plural |
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short |
τίθετε |
you put |
2 nd person plural |
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short |
τιθέᾱσι ( ν ) |
they put |
3 rd person plural |
Present Infinitive Active of τίθημι |
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short |
τιθέναι |
to put |
unmarked |
Readers of Euripides
of Athens,
Εὐριπίδης ὁ Ἀθηναῖος,
c. 480–406 BCE.
Readers of Euripides tend to view him in polarized ways. There are those who think that he criticized traditional religion and shocked contemporaries by representing mythical figures as everyday, unheroic people or even as abnormal or neurotic personalities. Others view him as conventionally religious. He is considered by some a misogynist because of his unsparing analysis of feminine passion. Others view him as highly sympathetic to the plight of women. Euripides’ enormous range spans these contradictory tendencies: he is both a rationalist and a romantic; he both criticizes the traditional gods and celebrates religion; some of his plays are more comic than tragic with happy endings, while others are bitter tragedies. He incorporates the new intellectual and scientific movements into his works but also conveys the irresistible power of the irrational. In modern times, Euripides has come to be increasingly appreciated for his intellectual subtlety, bold and original dramatic power, brilliant psychological insight, and ability to elicit unexpected symbolic meaning from ancient myth and cult.
Practice Translating δίδωμι. Translate the sentences below, which have been adapted from Euripides’ Medea ( Μήδεια ). Remember the meanings and functions of the cases presented in Module 7 . Nominative case endings are bolded, dative endings are highlighted, and accusative endings are underlined. Also in the sentences the definite article, which will be learned systematically in Modules 11 and 12 , is present in these forms: ἡ , τῇ , τό , τόν . Translate each one as the . Check your understanding with the translations in the Answer Key . Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
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Case |
Ending |
Function |
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Nominative |
- α , - η , - ος |
subject of the verb |
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Dative |
- ῃ , - ι , - ῳ |
indirect object of the verb |
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Accusative |
- α , - ην , - ν , - ο , - ον |
object of the verb |
- οὔκ , ἀλλὰ ἄλλ ος τ ὸν παῖδ α ἐκδίδωσιν αὐτ ῇ .
- ἡ γυν ὴ τῇ παιδ ὶ τιτθ όν δίδωσιν .
- τ ὸ ὄ χημ α ῞Ηλι ος δίδωσιν αὐτ ῇ .
- τ ὸν κόσμ ον ῞Ηλι ος δίδωσιν αὐτ ῇ .
- Φοῖ β ος αὐτ ῇ φάρμακ α δίδωσιν .
- καὶ οὐκ αὐτ ὴ αὐτ ὸν προδίδωσιν .
- α ὐτ ῇ διδόναι χάρι ν προθύμ η εἰμ ί .
- τ ῇ νύμφ ῃ διδόᾱσιν δῶρ α .
- καὶ αὐτ ὴν προδίδοτε .
- δίκ ην αὐτ ῇ καὶ αὐτ ῷ δίδως .
Verbs
* δίδωμι give ; δίκην δίδωμι I pay the penalty ; δίδωμι χάριν I give thanks
ἐκδίδωμι give up, surrender, give out (of one’s house), give (in marriage)
προδίδωμι betray, abandon, give up (to an enemy)
Adjectives, Nouns, Pronouns
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Νοminative Case |
Dative Case |
Accusative Case |
English Equivalent |
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* ἄλλος |
ἄλλῳ |
ἄλλον |
another, other |
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* αὐτή |
αὐτῇ |
αὐτήν |
she, her, hers |
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* αὐτός |
αὐτῷ |
αὐτόν |
he, him, his |
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* γυνή |
γυναικί |
γυναῖκα |
woman, wife |
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* δίκη |
δίκῃ |
δίκην |
justice, penalty |
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δῶρον |
δώρῳ |
δ ῶρον |
gift |
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῞Ηλιος |
Ἡλίῳ |
῞Ηλιον |
Helios |
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κόσμος |
κόσμῳ |
κόσμον |
ornament, dress |
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νύμφη |
νύμφῃ |
νύμφην |
bride |
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ὄ χημα |
ὀχήματι |
ὄχημα |
carriage, chariot |
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* παῖς |
παιδί |
παῖδα |
child |
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προθύμη |
προθύμῃ |
προθύμην |
eager |
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τιτθός |
τιτθῷ |
τιτθόν |
a woman’s breast |
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φάρμακα |
φαρμάκοις |
φάρμακα |
drugs, medicine |
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Φοῖβος |
Φοίβ ῳ |
Φοῖβον |
Phoibos Apollo |
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χάρις |
χάριτι |
χάριν |
grace, favor, thanks |
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* χρόνος |
χρόνῳ |
χρόνον |
time |
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* χρήματα |
χρήμασι ( ν ) |
χρήματα |
goods, money |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Translating τίθημι. Translate the sentences below, which have been adapted from Euripides’ Alkestis , ( Ἄλκηστις ). Remember the meanings and functions of the cases presented in Module 7 . Nominative case endings are bolded; genitive endings are italicized; dative endings are highlighted; and accusative endings are underlined. Also in the sentences the definite article, which will be learned systematically in Modules 11 and 12 , is present in this form: τούς . Translate it as the . Check your understanding with the translations in the Answer Key . Now go back and read each sentence two or three times, noticing with each rereading how much better your understanding of the sentence becomes. Make this a habit and you will improve quickly.
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Case |
Ending |
Function |
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Nominative |
- ηρ , - ι , - ον , - ος |
subject of the verb |
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Genitive |
- ου , - ων |
dependence; possession |
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Dative |
- ῃ , - ι , - ῳ |
indirect object of the verb; possession |
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Accusative |
- α , - ας , - ε , - ην , - ι , - ο , - ον , - ους |
object of preposition and verb |
- Ἄδμητος : ποῖ πόδ α τίθημι ; ποῖ ἵστημι ; τ ί λέγω ;
- Χορός : ἦ βαρυδαίμον α σ ὲ μήτ ηρ τίκτει .
- Ἄδμητος : Ἅιδ ου ἐπιθυμέω δώματ α ναίειν .
- Χορός : τ ὰ ἄνω κάτω τίθης καὶ ζηλόεις τ οὺς φθιμέν ους ;
- Ἄδμητος : γυναῖκ α Ἅιδ ῃ Θάνατ ος παραδίδωσιν .
- Χορός : σύ γε πάσχεις αἰαγμάτ ων ἄξι α .
- Ἄδμητος : ψυχ ὴν Ἅιδ ῃ ἐν χερσ ὶ τίθημι· οὐ γὰρ αὐγ ὰς θέλω προσοράειν .
- Χορός : δι ’ ὀδύν ας βαίνεις· σάφα τοῦτ ο οἶδα .
- Ἄδμητος : τ ί γὰρ ἀνδρ ί ἐστι κακ ὸν μεῖζ ον ἔχειν ;
- Χορός : θρόν ον τίθης ἐμο ὶ ἀλλὰ πέρ ας δέ γ ’ οὐκ ἀλγέ ων τίθης .
Adverbs and Verbs
|
βαίνεις you go |
ναίειν to inhabit |
|
ἐπιθυμέω I desire + inf. |
* οἶδα I know |
|
*ἔχειν to have |
*πάσχεις you suffer |
|
ζηλόεις you envy |
προσοράειν to look upon |
|
θέλω I wish + inf. |
σάφα clearly |
|
* ἵστημι stand, make stand , place |
τίκτει gives birth to |
|
*λέγω I say |
Adjectives, Nouns, Pronouns
|
Νοminative |
Genitive |
Dative |
Accusative |
English Equivalent |
|
Ἄδμητος |
Ἀδμήτου |
Ἀδμήτῳ |
Ἄδμητον |
Admetos |
|
αἰάγματα |
αἰαγμάτων |
αἰάγμασι ( ν ) |
αἰάγματα |
wails, cries |
|
Ἅιδης |
Ἅιδου |
Ἅιδῃ |
Ἅιδην |
Hades |
|
ἄ λγεα |
ἀλγέων |
ἄλγεσι ( ν ) |
ἄλγεα |
pain, grief |
|
* ἀνήρ |
ἀνδρός |
ἀνδρί |
ἄνδρα |
man, husband |
|
* ἄξια |
ἀξίων |
ἀξίοις |
ἄξια |
worthy things + gen. |
|
αὐγή |
αὐγῆς |
αὐγῇ |
αὐγήν |
sunlight |
|
βαρυ - δαίμων |
- δαίμονος |
- δαίμονι |
- δαίμονα |
luckless |
|
* γυνή |
γυναικός |
γυναικί |
γυναῖκα |
woman, wife |
|
δώματα |
δωμάτων |
δώμασι ( ν ) |
δώματα |
houses |
|
* ἐγώ |
ἐμοῦ |
ἐμοί |
ἐμέ |
I, me, mine |
|
Θάνατος |
Θανάτου |
Θανάτῳ |
Θάνατον |
Death |
|
θρόνος |
θρόνου |
θρόνῳ |
θρόνον |
seat, chair |
|
κακόν |
κακοῦ |
κακῷ |
κακόν |
bad, evil, cowardly |
|
μεῖζον |
μείζονος |
μείζονι |
μεῖζον |
greater |
|
* μήτηρ |
μητρός |
μητρί |
μητέρα |
mother |
|
ὀδύνη |
ὀδύνης |
ὀδύνῃ |
ὀδύνην |
pain, grief, distress |
|
οἱ φθί - μενοι |
τῶν - μένων |
τοῖς - μένοις |
τοὺς - μένους |
the dead |
|
πέρας |
πέρατος |
πέρατι |
πέρας |
end, limit, boundary |
|
πούς |
ποδός |
ποδί |
πόδα |
foot |
|
* σύ |
σοῦ |
σοί |
σέ |
you, you, yours |
|
τὰ ἄνω |
τῶν ἄνω |
τοῖς ἄνω |
τὰ ἄνω |
the things above |
|
τί |
τίνος |
τίνι |
τίνα |
who, what, which, why |
|
τοὺς φθιμένους see οἱ φθίμενοι |
||||
|
τοῦτο |
τούτου |
τούτῳ |
τοῦτο |
this, this thing |
|
Χορός |
Χοροῦ |
Χορῷ |
Χορόν |
Chorus-leader |
|
* χεῖ ρες |
χειρῶν |
χερσί ( ν ) |
χεῖρας |
hands; force, army |
|
* ψυχή |
ψυχῆς |
ψυχῇ |
ψυχήν |
soul, life |
- The asterisk indicates the top 250 most frequently occurring vocabulary, which you are to memorize.
Practice Parsing Greek Sentences. Parse each word of the sentences found below. For nouns and pronouns, give case and function. For verbs, give person, number, tense, mood, and voice. For adverbs and conjunctions, identify them. For prepositional phrases, give the preposition and the preposition’s object.
ἐγ ὼ μὲν ἔρομαι εἰ ἔνδον ἐστέ .
δίκην αὐτῇ καὶ αὐτῷ δίδως .
Check your answers with those in the Answer Key .
Module 9 Top 250 Vocabulary to be Memorized. Like learning the alphabet and endings, memorizing vocabulary is essential to acquiring language. The better you memorize the top 250 most frequently occurring vocabulary words, the greater mastery of the language you will have.
Verbs
δίδωμι give; allow ‘x’ in dat. or acc. + inf., ἐμὲ ( έμοί ) εὐτυχέειν δίδως you allow me to prosper
εἰμί be, be possible
ἵημι release, hurl, send ; (mid.) hasten
πάρειμι be near, be present ; (imper.) be possible
φημί say, affirm, assert


15: Laminar Flow
Smoothly flowing fluids, with the fluid flowing in undisrupted layers, are called laminar flows. There are several iconic laminar flows, whose velocity fields are readily found by solving the continuity and Navier-Stokes equations.
Plane Couette flow
Plane Couette flow consists of a fluid flowing between two infinite plates separated by a distance \(d\) . The lower plate is stationary, and the upper plate is moving to the right with velocity \(U\) . The pressure \(p\) is constant and the fluid is incompressible.
We look for a steady solution for the velocity field of the form
\[\nonumber \mathbf{u}(x, y, z)=(u(y), 0,0) . \nonumber \]
The incompressibility condition is automatically satisfied, and the first-component of the Navier-Stokes equation reduces to
\[\nonumber v \frac{\partial^{2} u}{\partial y^{2}}=0 \nonumber \]
Applying the boundary conditions \(u(0)=0\) and \(u(d)=U\) on the lower and upper plates, the laminar flow solution is given by
\[\nonumber u(y)=\frac{U y}{d} \nonumber \]
Channel flow
Channel flow, or Poiseuille flow, also consists of a fluid flowing between two infinite plates separated by a distance \(d\) , but with both plates stationary. Here, there is a constant pressure gradient along the \(x\) -direction in which the fluid flows. Again, we look for a steady solution for the velocity field of the form
\[\nonumber \mathbf{u}(x, y, z)=(u(y), 0,0) \nonumber \]
and with
\[\nonumber p=p(x) \nonumber \]
and
\[\frac{d p}{d x}=-G \nonumber \]
with \(G\) a positive constant. The first-component of the Navier-Stokes equation becomes
\[-\frac{1}{\rho} \frac{d p}{d x}+v \frac{d^{2} u}{d y^{2}}=0 . \nonumber \]
Using (15.1) in (15.2) leads to
\[\frac{d^{2} u}{d y^{2}}=-\frac{G}{v \rho} \nonumber \]
which can be solved using the no-slip boundary conditions \(u(0)=u(d)=0\) . We find
\[u(y)=\frac{G d^{2}}{2 v \rho}\left(\frac{y}{d}\right)\left(1-\frac{y}{d}\right) . \nonumber \]
The maximum velocity of the fluid occurs at the midline, \(y=d / 2\) , and is given by
\[u_{\max }=\frac{G d^{2}}{8 v \rho} \nonumber \]
Pipe flow
Pipe flow consists of flow through a pipe of circular cross-section radius \(R\) , with a constant pressure gradient along the pipe length. With the pressure gradient along the \(x\) -direction, we look for a steady solution of the velocity field of the the form
\[\mathbf{u}=(u(y, z), 0,0) . \nonumber \]
With the constant pressure gradient defined as in \((15.1)\) , the Navier-Stokes equation reduces to
\[\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=-\frac{G}{v \rho} . \nonumber \]
The use of polar coordinates in the \(y-z\) plane can aid in solving (15.7). With
\[u=u(r) \nonumber \]
we have
\[\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}=\frac{1}{r} \frac{d}{d r}\left(r \frac{d}{d r}\right), \nonumber \]
so that (15.7) becomes the differential equation
\[\frac{d}{d r}\left(r \frac{d u}{d r}\right)=-\frac{G r}{v \rho} \nonumber \]
with no-slip boundary condition \(u(R)=0 .\) The first integration from 0 to \(r\) yields
\[r \frac{d u}{d r}=-\frac{G r^{2}}{2 v \rho} \nonumber \]
and after division by \(r\) , the second integration from \(r\) to \(R\) yields
\[u(r)=\frac{G R^{2}}{4 v \rho}\left(1-\left(\frac{r}{R}\right)^{2}\right) \nonumber \]
The maximum velocity occurs at the pipe midline, \(r=0\) , and is given by
\[u_{\max }=\frac{G R^{2}}{4 v \rho} \nonumber \]


Magnetism
Skills to Develop
- Describe 2 types of dipoles
- Distinguish paramagnetism from diamagnetism
In 1923, Lewis wrote that the study of spectra and magnetism are the 2 best ways to learn about chemical bonding. Some types of spectra are discussed here . In this section, we will describe the magnetic evidence Lewis used. You are familiar with normal magnets, like the ones used to stick restaurant menus to fridges. In this section, however, we are more interested in the magnetic properties of molecular materials, rather than metallic or semi-ionic solids. Most materials are non-magnetic. The molecular materials that are magnetic still wouldn't stick to your fridge, because they only really act magnetic in a magnetic field. Normally, each molecular magnetic has a random direction because the interactions between the molecules are weak. (Just like how molecular materials melt at lower temperature than metals or rocks). If you put them in a magnetic field, if they are cold enough, they will start to line up with the field. If they are too hot, they will continue to move randomly. This is called paramagnetism .
Dipoles
Let's start by reviewing electric dipoles. An electric dipole is something that had separated electrical charges. For instance, an HF molecule or a water molecule have an electric dipole moment defined as
\[\mu = q \times d\]
where q is the partial charge on each end of the molecule and d is the distance between the charges.
Generally, solvents with large dipoles have high dielectric constants . In Coulomb's law, the dielectric constant D reduces the force between charges.
\[F= \frac{kQq}{Dr^{2}}\]
Bigger D means less force. This explains how ionic substances dissolve in solvents with large dipole moments and large D. When the ions separate and have water between them, they no longer attract each other strongly.
Although you can have a point charge, magnets are always dipoles (have a North and South pole). Just like solvents and other materials have dielectric constants, they also have magnetic permeabilities.
Magnetic Measurements
How do you measure magnetic moments or magnetic dipoles in molecules? The old way is to put the molecular material between the poles of a big magnet and see what forces are present. For instance, if you put a paramagnetic material (that is cold enough) between the N and S magnetic poles, it will line up with the field and be attracted by the field. Thus if you weigh a paramagnetic material in a magnetic field, it will be heavier than without the field. Most materials are diamagnetic , and get lighter in magnetic fields, because they are repelled by the field. So the easiest way to see if a material is magnetic is to weigh it with and without a magnetic field. (Now there is a fancier method also, called SQUID, which is too complicated to explain here.)
How is Magnetism Related to Chemical Bonding?
Lewis correctly realized that molecules with an odd number of electrons (such as NO, with 7+8 =15, but not CO, with 6+8 =14) have some unusual properties. They are paramagnetic, usually strongly colored, and quite reactive. The huge majority of molecules have even numbers of electrons, and are diamagnetic. Thus, paramagnetism seems to be a result of unpaired electrons, while diamagnetism is a result of paired electrons. Transition metal compounds are often colored and magnetic; most compounds of C,H,O,N etc (such as in your body) have little color, are diamagnetic, and not so reactive. Basically, almost all molecules have all their electrons in pairs.


14.3: The Accounting Profession
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2. What are the differences between public and private accountants, and how has federal legislation affected their work?
When you think of accountants, do you picture someone who works in a back room, hunched over a desk, wearing a green eye shade and scrutinizing pages and pages of numbers? Although today’s accountants still must love working with numbers, they now work closely with their clients to not only prepare financial reports but also help them develop good financial practices. Advances in technology have taken the tedium out of the number-crunching and data-gathering parts of the job and now offer powerful analytical tools as well. Therefore, accountants must keep up with information technology trends. The accounting profession has grown due to the increased complexity, size, and number of businesses and the frequent changes in the tax laws. Accounting is now a $95 billion-plus industry. The more than 1.4 million accountants in the United States are classified as either public accountants or private (corporate) accountants. They work in public accounting firms, private industry, education, and government, and about 10 percent are self-employed. The job outlook for accountants over the next decade is positive; the Bureau of Labor Statistics projects that accounting and auditing jobs will increase 11 percent faster than many other industries in the U.S. economy. 2
Public Accountants
Independent accountants who serve organizations and individuals on a fee basis are called public accountants . Public accountants offer a wide range of services, including preparation of financial statements and tax returns, independent auditing of financial records and accounting methods, and management consulting. Auditing , the process of reviewing the records used to prepare financial statements, is an important responsibility of public accountants. They issue a formal auditor’s opinion indicating whether the statements have been prepared in accordance with accepted accounting rules. This written opinion is an important part of a company’s annual report.
The largest public accounting firms, called the Big Four, operate worldwide and offer a variety of business consulting services in addition to accounting services. In order of size, they are Deloitte, PwC (PricewaterhouseCoopers), EY (Ernst & Young), and KPMG International. 3 A former member of this group, Arthur Andersen, disbanded in 2002 as a result of the Enron scandal.
To become a certified public accountant (CPA) , an accountant must complete an approved bachelor’s degree program and pass a test prepared by the American Institute of CPAs (AICPA). Each state also has requirements for CPAs, such as several years’ on-the-job experience and continuing education. Only CPAs can issue the auditor’s opinion on a firm’s financial statements. Most CPAs first work for public accounting firms and later may become private accountants or financial managers. Of the more than 418,000 accountants who belong to the AICPA, 47 percent work in public accounting firms and 39 percent in business and industry. 4
Private Accountants
Accountants employed to serve one particular organization are private accountants . Their activities include preparing financial statements, auditing company records to be sure employees follow accounting policies and procedures, developing accounting systems, preparing tax returns, and providing financial information for management decision-making. Whereas some private accountants hold the CPA designation, managerial accountants also have a professional certification program. Requirements to become a certified management accountant (CMA) include passing an examination.
Reshaping the Accounting Environment
Although our attention was focused on big-name accounting scandals in the late 1990s and early 2000s, an epidemic of accounting irregularities was also taking place in the wider corporate arena. The number of companies restating annual financial statements grew at an alarming rate, tripling from 1997 to 2002. In the wake of the numerous corporate financial scandals, Congress and the accounting profession took major steps to prevent future accounting irregularities. These measures targeted the basic ways, cited by a report from the AICPA, that companies massaged financial reports through creative, aggressive, or inappropriate accounting techniques, including:
- Committing fraudulent financial reporting
- Stretching accounting rules to significantly enhance financial results
- Following appropriate accounting rules but using loopholes to manage financial results
Why did companies willfully push accounting to the edge—and over it—to artificially pump up revenues and profits? Looking at the companies involved in the scandals, some basic similarities have emerged:
- A company culture of arrogance and above-average tolerance for risk
- Interpretation of accounting policies to their advantage and manipulation of the rules to get to a predetermined result and conceal negative financial information
- Compensation packages tied to financial or operating targets, making executives and managers greedy and pressuring them to find sometimes-questionable ways to meet what may have been overly optimistic goals
- Ineffective checks and balances, such as audit committees, boards of directors, and financial control procedures, that were not independent from management
- Centralized financial reporting that was tightly controlled by top management, increasing the opportunity for fraud
- Financial performance benchmarks that were often out of line with the companies’ industry
- Complicated business structures that clouded how the company made its profits
- Cash flow from operations that seemed out of line with reported earnings (You’ll learn about this important difference between cash and reported earnings in the sections on the income statement and statement of cash flows.)
- Acquisitions made quickly, often to show growth rather than for sound business reasons; management focused more on buying new companies than making the existing operations more profitable 5
Companies focused on making themselves look good in the short term, doing whatever was necessary to top past performance and to meet the expectations of investment analysts, who project earnings, and investors, who panic when a company misses the analysts’ forecasts. Executives who benefited when stock prices rose had no incentive to question the earnings increases that led to the price gains.
These number games raised serious concerns about the quality of earnings and questions about the validity of financial reports. Investors discovered to their dismay that they could neither assume that auditors were adequately monitoring their clients’ accounting methods nor depend on the integrity of published financial information.
Better Numbers Ahead
Over the past 15 years, a number of accounting reforms have been put in place to set better standards for accounting, auditing, and financial reporting. Investors, now aware of the possibility of various accounting shenanigans, are avoiding companies that use complicated financial structures and off-the-books financing.
In 2002, the Sarbanes-Oxley Act (commonly referred to as SOX) went into effect. This law, one of the most extensive pieces of business legislation passed by Congress, was designed to address the investing public’s lack of trust in corporate America. It redefines the public corporation–auditor relationship and restricts the types of services auditors can provide to clients. The Act clarifies auditor-independence issues, places increased accountability on a company’s senior executives and management, strengthens disclosure of insider transactions (an employee selling stock based on information not known by the public), and prohibits loans to executives.
An independent five-member Public Company Accounting Oversight Board (PCAOB) was given the authority to set and amend auditing, quality control, ethics, independence, and other standards for audit reports. The Act specifies that all PCAOB members be financially literate. Two members must have their CPA designation, and the other three cannot be or have been CPAs. Appointed and overseen by the Securities and Exchange Commission (SEC), the PCAOB can also inspect accounting firms; investigate breaches of securities law, standards, competency, and conduct; and take disciplinary action. The corporate Board registers public accounting firms, as the Act now requires. Altering or destroying key audit documents now carries felony charges and increased penalties.
Other key provisions of the Act cover the following areas:
- Auditing standards: The Board must include in its standards several requirements, such as maintaining audit work papers and other documentation for audit reports for seven years, the review and approval of audit reports by a second partner, and audit standards for quality control and review of internal control procedures.
- Financial disclosure: Companies must clearly disclose all transactions that may have a material current or future effect on their financial condition, including those that are off the books or with unconsolidated entities (related companies whose results the company is not required to combine with its own financial statements under current accounting rules). Management and major stockholders must disclose transactions such as sales of company stock within two days of the transaction. The company must disclose its code of ethics for senior financial executives. Any significant changes in a company’s operations or financial condition must be disclosed “on a rapid and current basis.”
- Financial statement certification: Chief executive officers and chief financial officers must certify company financial statements, with severe criminal and civil penalties for false certification. If securities fraud results in restatement of financial reports, these executives will lose any stock-related profits and bonuses they received prior to the restatement.
- Internal controls: Each company must have appropriate internal control procedures in place for financial reporting, and its annual report must include a report on implementation of those controls to assure the integrity of financial reports.
- Consulting work: The Act restricts the non-auditing work auditors may perform for a client. In the past, the large accounting firms had expanded their role to include a wide range of advisory services that went beyond their traditional task of validating a company’s financial information. Conflicts of interest arose when the same firm earned lucrative fees for both audit and consulting work for the same client. 6
Other regulatory organizations also took steps to prevent future abuses. In September 2002, the AICPA Auditing Standards Board(ASB) issued expanded guidelines to help auditors uncover fraud while conducting audits. The New York Stock Exchange stiffened its listing requirements so that the majority of directors at listed companies must be independent and not employees of the corporation. Nor can auditors serve on clients’ boards for five years. Companies listed in the Nasdaq marketplace cannot hire former auditors at any level for three years.
In response to the passage of Sarbanes-Oxley and other regulations, companies implemented new control measures and improved existing ones. The burdens in both cost and time have been considerable. Many companies had to redesign and restructure financial systems to improve efficiency. Some finance executives believe that their investment in increased controls has improved shareholder perceptions of their company’s ethics. Others, however, reported that costs depressed earnings and negatively affected stock prices. Despite the changes and costs associated with SOX compliance, 15 years after the law’s implementation, many business executives believe that the process has helped them fine-tune financial activities and reporting while addressing dynamic changes in the market and other economic challenges. 7
CONCEPT CHECK
- Compare the responsibilities of public and private accountants. How are they certified?
- Summarize the major changes affecting accounting and corporate reporting and the reasons for them.


In this section, we will explain the importance of listening.
The Importance of Listening
If you were to break up the various types of communication you use throughout your day, listening would make up about 42 – 60 percent. Problematically, many people engage in passive versus active listening, which can cause miscommunication and have a negative effect on relationships. Listening is an important skill to cultivate, both personally and professionally. In this chapter, we will explain the importance of listening, the stages of the listening process, listening styles, some common listening barriers, and ways to improve our listening skills.
Learning Objectives
How does the listening process work?
What are some of the challenges to effective listening?
How can we improve our listening, and why might it be important to do so?
Successful students will be able to:
explain the importance of listening
differentiate between five stages of the listening process
identify four listening styles
describe six barriers to effective listening
demonstrate ways to improve listening in all stages of the process
recognize contextual influences on listening
recognize the value of reflection on listening and relationships
Sections 7.0: Interpersonal Communication Abridged Textbook (I.C.A.T.); Central New Mexico Community College; 2019;
CC BY NC SA 4.0
Sections 7.1.1 – 7.5.3: adapted from Communication in the Real World: An Introduction to Communication Studies; University of Minnesota; 2016;
CC BY NC SA 4.0


2.2: Concentration Units - Expressing Solute Proportions
- To describe the concentration of a solution in the way that is most appropriate for a particular problem or application.
- To be familiar with the different units used to express the concentrations of a solution.
There are several different ways to quantitatively describe the concentration of a solution. For example, molarity is a useful way to describe solution concentrations for reactions that are carried out in solution. Mole fractions are used not only to describe gas concentrations but also to determine the vapor pressures of mixtures of similar liquids. Example \(\PageIndex{1}\) reviews the methods for calculating the molarity and mole fraction of a solution when the masses of its components are known.
Commercial vinegar is essentially a solution of acetic acid in water. A bottle of vinegar has 3.78 g of acetic acid per 100.0 g of solution. Assume that the density of the solution is 1.00 g/mL.
- What is its molarity?
- What is its mole fraction?
Given : mass of substance and mass and density of solution
Asked for : molarity and mole fraction
Strategy :
- Calculate the number of moles of acetic acid in the sample. Then calculate the number of liters of solution from its mass and density. Use these results to determine the molarity of the solution.
- Determine the mass of the water in the sample and calculate the number of moles of water. Then determine the mole fraction of acetic acid by dividing the number of moles of acetic acid by the total number of moles of substances in the sample.
Solution :
A: The molarity is the number of moles of acetic acid per liter of solution. We can calculate the number of moles of acetic acid as its mass divided by its molar mass.
\[ \begin{align*} \text{moles } \ce{CH_3CO_2H} &=\dfrac{3.78\; \cancel{\ce{g}}\; \ce{CH_3CO_2H}}{60.05\; \cancel{\ce{g}}/\ce{mol}} \\[4pt] &=0.0629 \; \ce{mol} \end{align*} \nonumber \]
The volume of the solution equals its mass divided by its density.
\[ \begin{align*} \text{volume} &=\dfrac{\text{mass}}{\text{density}} \\[4pt] &=\dfrac{100.0\; \cancel{\ce{g}}\; \text{solution}}{1.00\; \cancel{\ce{g}}/\ce{mL}}=100\; mL\nonumber \end{align*} \nonumber \]
Then calculate the molarity directly.
\[ \begin{align*} \text{molarity of } \ce{CH_3CO_2H} &=\dfrac{\text{moles } \ce{CH3CO2H} }{\text{liter solution}} \\[4pt] &=\dfrac{0.0629\; mol\; \ce{CH_3CO_2H}}{(100\; \cancel{\ce{mL}})(1\; L/1000\; \cancel{\ce{mL}})}=0.629\; M \; \ce{CH_3CO_2H} \end{align*} \nonumber \]
This result makes intuitive sense. If 100.0 g of aqueous solution (equal to 100 mL) contains 3.78 g of acetic acid, then 1 L of solution will contain 37.8 g of acetic acid, which is a little more than \(\ce{ 1/2}\) mole. Keep in mind, though, that the mass and volume of a solution are related by its density; concentrated aqueous solutions often have densities greater than 1.00 g/mL.
B: To calculate the mole fraction of acetic acid in the solution, we need to know the number of moles of both acetic acid and water. The number of moles of acetic acid is 0.0629 mol, as calculated in part (a). We know that 100.0 g of vinegar contains 3.78 g of acetic acid; hence the solution also contains (100.0 g − 3.78 g) = 96.2 g of water. We have
\[moles\; \ce{H_2O}=\dfrac{96.2\; \cancel{\ce{g}}\; \ce{H_2O}}{18.02\; \cancel{\ce{g}}/mol}=5.34\; mol\; \ce{H_2O}\nonumber \]
The mole fraction \(\chi\) of acetic acid is the ratio of the number of moles of acetic acid to the total number of moles of substances present:
\[ \begin{align*} \chi_{\ce{CH3CO2H}} &=\dfrac{moles\; \ce{CH_3CO_2H}}{moles \; \ce{CH_3CO_2H} + moles\; \ce{H_2O}} \\[4pt] &=\dfrac{0.0629\; mol}{0.0629 \;mol + 5.34\; mol} \\[4pt] &=0.0116=1.16 \times 10^{−2} \end{align*} \nonumber \]
This answer makes sense, too. There are approximately 100 times as many moles of water as moles of acetic acid, so the ratio should be approximately 0.01.
A solution of \(\ce{HCl}\) gas dissolved in water (sold commercially as “muriatic acid,” a solution used to clean masonry surfaces) has 20.22 g of \(\ce{HCl}\) per 100.0 g of solution, and its density is 1.10 g/mL.
- What is its molarity?
- What is its mole fraction?
- Answer a
-
6.10 M HCl
- Answer b
-
\(\chi_{HCl} = 0.111\)
The concentration of a solution can also be described by its molality (m), the number of moles of solute per kilogram of solvent:
\[ \text{molality (m)} =\dfrac{\text{moles solute}}{\text{kilogram solvent}} \label{Eq1} \]
Molality, therefore, has the same numerator as molarity (the number of moles of solute) but a different denominator (kilogram of solvent rather than liter of solution). For dilute aqueous solutions, the molality and molarity are nearly the same because dilute solutions are mostly solvent. Thus because the density of water under standard conditions is very close to 1.0 g/mL, the volume of 1.0 kg of \(H_2O\) under these conditions is very close to 1.0 L, and a 0.50 M solution of \(KBr\) in water, for example, has approximately the same concentration as a 0.50 m solution.
Another common way of describing concentration is as the ratio of the mass of the solute to the total mass of the solution. The result can be expressed as mass percentage, parts per million (ppm), or parts per billion (ppb):
\[ \begin{align} \text{mass percentage}&=\dfrac{\text{mass of solute}}{\text{mass of solution}} \times 100 \label{Eq2} \\[4pt] \text{parts per million (ppm)} &=\dfrac{\text{mass of solute}}{\text{mass of solution}} \times 10^{6} \label{Eq3} \\[4pt] \text{parts per billion (ppb)}&=\dfrac{\text{mass of solute}}{\text{mass of solution}} \times 10^{9} \label{Eq4} \end{align} \]
In the health sciences, the concentration of a solution is often expressed as parts per thousand (ppt), indicated as a proportion. For example, adrenalin, the hormone produced in high-stress situations, is available in a 1:1000 solution, or one gram of adrenalin per 1000 g of solution.
The labels on bottles of commercial reagents often describe the contents in terms of mass percentage. Sulfuric acid, for example, is sold as a 95% aqueous solution, or 95 g of \(\ce{H_2SO_4}\) per 100 g of solution. Parts per million and parts per billion are used to describe concentrations of highly dilute solutions. These measurements correspond to milligrams and micrograms of solute per kilogram of solution, respectively. For dilute aqueous solutions, this is equal to milligrams and micrograms of solute per liter of solution (assuming a density of 1.0 g/mL).
Several years ago, millions of bottles of mineral water were contaminated with benzene at ppm levels. This incident received a great deal of attention because the lethal concentration of benzene in rats is 3.8 ppm. A 250 mL sample of mineral water has 12.7 ppm of benzene. Because the contaminated mineral water is a very dilute aqueous solution, we can assume that its density is approximately 1.00 g/mL.
- What is the molarity of the solution?
- What is the mass of benzene in the sample?
Given : volume of sample, solute concentration, and density of solution
Asked for : molarity of solute and mass of solute in 250 mL
Strategy :
- Use the concentration of the solute in parts per million to calculate the molarity.
- Use the concentration of the solute in parts per million to calculate the mass of the solute in the specified volume of solution.
Solution :
a. A To calculate the molarity of benzene, we need to determine the number of moles of benzene in 1 L of solution. We know that the solution contains 12.7 ppm of benzene. Because 12.7 ppm is equivalent to 12.7 mg/1000 g of solution and the density of the solution is 1.00 g/mL, the solution contains 12.7 mg of benzene per liter (1000 mL). The molarity is therefore
\[\begin{align*} \text{molarity}&=\dfrac{\text{moles}}{\text{liter solution}} \\[4pt] &=\dfrac{(12.7\; \cancel{mg}) \left(\frac{1\; \cancel{g}}{1000\; \cancel{mg}}\right)\left(\frac{1\; mol}{78.114\; \cancel{g}}\right)}{1.00\; L} \\[4pt] &=1.63 \times 10^{-4} M\end{align*} \nonumber \]
b. B We are given that there are 12.7 mg of benzene per 1000 g of solution, which is equal to 12.7 mg/L of solution. Hence the mass of benzene in 250 mL (250 g) of solution is
\[\begin{align*} \text{mass of benzene} &=\dfrac{(12.7\; mg\; \text{benzene})(250\; \cancel{mL})}{1000\; \cancel{mL}} \\[4pt] &=3.18\; mg \\[4pt] &=3.18 \times 10^{-3}\; g\; \text{benzene} \end{align*} \nonumber \]
The maximum allowable concentration of lead in drinking water is 9.0 ppb.
- What is the molarity of \(\ce{Pb^{2+}}\) in a 9.0 ppb aqueous solution?
- Use your calculated concentration to determine how many grams of \(\ce{Pb^{2+}}\) are in an 8 oz glass of water.
- Answer a
-
4.3 × 10 −8 M
- Answer b
-
2 × 10 −6 g
How do chemists decide which units of concentration to use for a particular application? Although molarity is commonly used to express concentrations for reactions in solution or for titrations, it does have one drawback—molarity is the number of moles of solute divided by the volume of the solution, and the volume of a solution depends on its density, which is a function of temperature. Because volumetric glassware is calibrated at a particular temperature, typically 20°C, the molarity may differ from the original value by several percent if a solution is prepared or used at a significantly different temperature, such as 40°C or 0°C. For many applications this may not be a problem, but for precise work these errors can become important. In contrast, mole fraction, molality, and mass percentage depend on only the masses of the solute and solvent, which are independent of temperature.
Mole fraction is not very useful for experiments that involve quantitative reactions, but it is convenient for calculating the partial pressure of gases in mixtures, as discussed previously. Mole fractions are also useful for calculating the vapor pressures of certain types of solutions. Molality is particularly useful for determining how properties such as the freezing or boiling point of a solution vary with solute concentration. Because mass percentage and parts per million or billion are simply different ways of expressing the ratio of the mass of a solute to the mass of the solution, they enable us to express the concentration of a substance even when the molecular mass of the substance is unknown. Units of ppb or ppm are also used to express very low concentrations, such as those of residual impurities in foods or of pollutants in environmental studies.
Table \(\PageIndex{1}\) summarizes the different units of concentration and typical applications for each. When the molar mass of the solute and the density of the solution are known, it becomes relatively easy with practice to convert among the units of concentration we have discussed, as illustrated in Example \(\PageIndex{3}\).
| Unit | Definition | Application |
|---|---|---|
| *The molarity of a solution is temperature dependent, but the other units shown in this table are independent of temperature. | ||
| molarity (M) | moles of solute/liter of solution (mol/L) | Used for quantitative reactions in solution and titrations; mass and molecular mass of solute and volume of solution are known. |
| mole fraction (\(\chi\)) | moles of solute/total moles present (mol/mol) | Used for partial pressures of gases and vapor pressures of some solutions; mass and molecular mass of each component are known. |
| molality (m) | moles of solute/kg of solvent (mol/kg) | Used in determining how colligative properties vary with solute concentration; masses and molecular mass of solute are known. |
| mass percentage (%) | [mass of solute (g)/mass of solution (g)] × 100 | Useful when masses are known but molecular masses are unknown. |
| parts per thousand (ppt) | [mass of solute/mass of solution] × 10 3 (g solute/kg solution) | Used in the health sciences, ratio solutions are typically expressed as a proportion, such as 1:1000. |
| parts per million (ppm) | [mass of solute/mass of solution] × 10 6 (mg solute/kg solution) | Used for trace quantities; masses are known but molecular masses may be unknown. |
| parts per billion (ppb) | [mass of solute/mass of solution] × 10 9 (µg solute/kg solution) | Used for trace quantities; masses are known but molecular masses may be unknown. |
Vodka is essentially a solution of ethanol in water. Typical vodka is sold as “80 proof,” which means that it contains 40.0% ethanol by volume. The density of pure ethanol is 0.789 g/mL at 20°C. If we assume that the volume of the solution is the sum of the volumes of the components (which is not strictly correct), calculate the following for the ethanol in 80-proof vodka.
- the mass percentage
- the mole fraction
- the molarity
- the molality
Given : volume percent and density
Asked for : mass percentage, mole fraction, molarity, and molality
Strategy :
- Use the density of the solute to calculate the mass of the solute in 100.0 mL of solution. Calculate the mass of water in 100.0 mL of solution.
- Determine the mass percentage of solute by dividing the mass of ethanol by the mass of the solution and multiplying by 100.
- Convert grams of solute and solvent to moles of solute and solvent. Calculate the mole fraction of solute by dividing the moles of solute by the total number of moles of substances present in solution.
- Calculate the molarity of the solution: moles of solute per liter of solution. Determine the molality of the solution by dividing the number of moles of solute by the kilograms of solvent.
Solution :
The key to this problem is to use the density of pure ethanol to determine the mass of ethanol (\(CH_3CH_2OH\)), abbreviated as EtOH, in a given volume of solution. We can then calculate the number of moles of ethanol and the concentration of ethanol in any of the required units. A Because we are given a percentage by volume, we assume that we have 100.0 mL of solution. The volume of ethanol will thus be 40.0% of 100.0 mL, or 40.0 mL of ethanol, and the volume of water will be 60.0% of 100.0 mL, or 60.0 mL of water. The mass of ethanol is obtained from its density:
\[mass\; of\; EtOH=(40.0\; \cancel{mL})\left(\dfrac{0.789\; g}{\cancel{mL}}\right)=31.6\; g\; EtOH\nonumber \]
If we assume the density of water is 1.00 g/mL, the mass of water is 60.0 g. We now have all the information we need to calculate the concentration of ethanol in the solution.
B The mass percentage of ethanol is the ratio of the mass of ethanol to the total mass of the solution, expressed as a percentage:
\[ \begin{align*} \%EtOH &=\left(\dfrac{mass\; of\; EtOH}{mass\; of\; solution}\right)(100) \\[4pt] &=\left(\dfrac{31.6\; \cancel{g}\; EtOH}{31.6\; \cancel{g} \;EtOH +60.0\; \cancel{g} \; H_2O} \right)(100) \\[4pt]&= 34.5\%\end{align*} \nonumber \]
C The mole fraction of ethanol is the ratio of the number of moles of ethanol to the total number of moles of substances in the solution. Because 40.0 mL of ethanol has a mass of 31.6 g, we can use the molar mass of ethanol (46.07 g/mol) to determine the number of moles of ethanol in 40.0 mL:
\[ \begin{align*} moles\; \ce{CH_3CH_2OH}&=(31.6\; \cancel{g\; \ce{CH_3CH_2OH}}) \left(\dfrac{1\; mol}{46.07\; \cancel{g\; \ce{CH_3CH_2OH}}}\right) \\[4pt] &=0.686 \;mol\; \ce{CH_3CH_2OH} \end{align*} \nonumber \]
Similarly, the number of moles of water is
\[ moles \;\ce{H_2O}=(60.0\; \cancel{g \; \ce{H_2O}}) \left(\dfrac{1 \;mol\; \ce{H_2O}}{18.02\; \cancel{g\; \ce{H_2O}}}\right)=3.33\; mol\; \ce{H_2O}\nonumber \]
The mole fraction of ethanol is thus
\[ \chi_{\ce{CH_3CH_2OH}}=\dfrac{0.686\; \cancel{mol}}{0.686\; \cancel{mol} + 3.33\;\cancel{ mol}}=0.171\nonumber \]
D The molarity of the solution is the number of moles of ethanol per liter of solution. We already know the number of moles of ethanol per 100.0 mL of solution, so the molarity is
The molality of the solution is the number of moles of ethanol per kilogram of solvent. Because we know the number of moles of ethanol in 60.0 g of water, the calculation is again straightforward:
\[ m_{\ce{CH_3CH_2OH}}=\left(\dfrac{0.686\; mol\; EtOH}{60.0\; \cancel{g}\; H_2O } \right) \left(\dfrac{1000\; \cancel{g}}{kg}\right)=\dfrac{11.4\; mol\; EtOH}{kg\; H_2O}=11.4 \;m\nonumber \]
A solution is prepared by mixing 100.0 mL of toluene with 300.0 mL of benzene. The densities of toluene and benzene are 0.867 g/mL and 0.874 g/mL, respectively. Assume that the volume of the solution is the sum of the volumes of the components. Calculate the following for toluene.
- mass percentage
- mole fraction
- molarity
- molality
- Answer a
-
mass percentage toluene = 24.8%
- Answer b
-
\(\chi_{toluene} = 0.219\)
- Answer c
-
2.35 M toluene
- Answer d
-
3.59 m toluene
A Video Discussing Different Measures of Concentration. Video Link: Measures of Concentration, YouTube (opens in new window) [youtu.be]
A Video Discussing how to Convert Measures of Concentration. Video Link: Converting Units of Concentration, YouTube(opens in new window) [youtu.be]
Summary
Different units are used to express the concentrations of a solution depending on the application. The concentration of a solution is the quantity of solute in a given quantity of solution. It can be expressed in several ways: molarity (moles of solute per liter of solution); mole fraction, the ratio of the number of moles of solute to the total number of moles of substances present; mass percentage, the ratio of the mass of the solute to the mass of the solution times 100; parts per thousand (ppt), grams of solute per kilogram of solution; parts per million (ppm), milligrams of solute per kilogram of solution; parts per billion (ppb), micrograms of solute per kilogram of solution; and molality (m), the number of moles of solute per kilogram of solvent.