3.2 Section • Determining the Formula of an Unknown Compound A-1 Common Mathematical Operations in Chemistry I n addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry: manipulating logarithms, using exponential nota- tion, solving quadratic equations, and graphing data. Each is discussed briefly below. MANIPULATING LOGARITHMS Meaning and Properties of Logarithms A logarithm is an exponent. Specifically, if x n A, we can say that the logarithm to the base x of the number A is n, and we can denote it as log x A n Because logarithms are exponents, they have the following properties: log x 1 0 log x (A 3 B) log x A 1 log x B log x A B 5 log x A 2 log x B log x A y y log x A Types of Logarithms Common and natural logarithms are used in chemistry and the other sciences. 1. For common logarithms, the base (x in the examples above) is 10, but they are written without specifying the base; that is, log 10 A is written more simply as log A; thus, the notation log means base 10. The common logarithm of 1000 is 3; in other words, you must raise 10 to the 3rd power to obtain 1000: log 1000 3 or 10 3 1000 Similarly, we have log 10 1 or 10 1 10 log 1,000,000 6 or 10 6 1,000,000 log 0.001 23 or 10 23 0.001 log 853 2.931 or 10 2.931 853 The last example illustrates an important point about significant figures with all loga- rithms: the number of significant figures in the number equals the number of digits to the right of the decimal point in the logarithm. That is, the number 853 has three significant figures, and the logarithm 2.931 has three digits to the right of the decimal point. To find a common logarithm with an electronic calculator, enter the number and press the log button. 2. For natural logarithms, the base is the number e, which is 2.71828 . . . , and log e A is written ln A; thus, the notation ln means base e. The relationship between the common and natural logarithms is easily obtained: log 10 1 and ln 10 2.303 Appendix A A-1
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3.2 Section • Determining the Formula of an Unknown Compound A-1
Common Mathematical Operations in Chemistry
In addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry: manipulating logarithms, using exponential nota-
tion, solving quadratic equations, and graphing data. Each is discussed briefly below.
ManIpUlatIng logarIthMsMeaning and Properties of LogarithmsA logarithm is an exponent. Specifically, if xn A, we can say that the logarithm to the base x of the number A is n, and we can denote it as
logx A n
Because logarithms are exponents, they have the following properties:
logx 1 0
logx (A 3 B) logx A 1 logx B
logx A
B5 logx A 2 logx B
logx Ay y logx A
Types of LogarithmsCommon and natural logarithms are used in chemistry and the other sciences.
1. For common logarithms, the base (x in the examples above) is 10, but they are written without specifying the base; that is, log10 A is written more simply as log A; thus, the notation log means base 10. The common logarithm of 1000 is 3; in other words, you must raise 10 to the 3rd power to obtain 1000:
log 1000 3 or 103 1000
Similarly, we have
log 10 1 or 101 10 log 1,000,000 6 or 106 1,000,000 log 0.001 23 or 1023 0.001 log 853 2.931 or 102.931 853
The last example illustrates an important point about significant figures with all loga-rithms: the number of significant figures in the number equals the number of digits to the right of the decimal point in the logarithm. That is, the number 853 has three significant figures, and the logarithm 2.931 has three digits to the right of the decimal point. To find a common logarithm with an electronic calculator, enter the number and press the log button.
2. For natural logarithms, the base is the number e, which is 2.71828 . . . , and loge A is written ln A; thus, the notation ln means base e. The relationship between the common and natural logarithms is easily obtained:
log 10 1 and ln 10 2.303
Appendix A
A-1
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A-2 Appendix A • Common Mathematical Operations in Chemistry
Therefore, we have
ln A 5 2.303 log A
To find a natural logarithm with an electronic calculator, enter the number and press the ln button. If your calculator does not have an ln button, enter the number, press the log button, and multiply by 2.303.
AntilogarithmsThe antilogarithm is the base raised to the logarithm:
antilogarithm (antilog) of n is 10n
Using two of the earlier examples, the antilog of 3 is 1000, and the antilog of 2.931 is 853. To obtain the antilog with a calculator, enter the number and press the 10x button. Similarly, to obtain the natural antilogarithm, enter the number and press the ex button. [On some calculators, enter the number and first press inv and then the log (or ln) button.]
Using ExponEntiAl (sciEntific) notAtionMany quantities in chemistry are very large or very small. For example, in the conven-tional way of writing numbers, the number of gold atoms in 1 gram of gold is
59,060,000,000,000,000,000,000 atoms (to four significant figures)
As another example, the mass in grams of one gold atom is
0.0000000000000000000003272 g (to four significant figures)
Exponential (scientific) notation provides a much more practical way of writing such numbers. In exponential notation, we express numbers in the form
A10n
where A (the coefficient) is greater than or equal to 1 and less than 10 (that is, 1 A 10), and n (the exponent) is an integer. If the number we want to express in exponential notation is larger than 1, the exponent is positive (n 0); if the number is smaller than 1, the exponent is negative (n 0). The size of n tells the number of places the decimal point (in conventional notation) must be moved to obtain a coefficient A greater than or equal to 1 and less than 10 (in exponential notation). In exponential notation, 1 gram of gold contains 5.9061022 atoms, and each gold atom has a mass of 3.2721022 g.
Changing Between Conventional and Exponential NotationIn order to use exponential notation, you must be able to convert to it from conven-tional notation, and vice versa.
1. To change a number from conventional to exponential notation, move the decimal point to the left for numbers equal to or greater than 10 and to the right for num-bers between 0 and 1:
75,000,000 changes to 7.5107 (decimal point 7 places to the left)0.006042 changes to 6.042103 (decimal point 3 places to the right)
2. To change a number from exponential to conventional notation, move the deci-mal point the number of places indicated by the exponent to the right for num-bers with positive exponents and to the left for numbers with negative exponents:
1.38105 changes to 138,000 (decimal point 5 places to the right)8.41106 changes to 0.00000841 (decimal point 6 places to the left)
Appendix A • Common Mathematical Operations in Chemistry A-3
3. An exponential number with a coefficient greater than 10 or less than 1 can be changed to the standard exponential form by converting the coefficient to the stan-dard form and adding the exponents:
582.3106 changes to 5.823 102 106 5 5.82310(26) 5 5.823108
0.0043104 changes to 4.3 103 104 5 4.310[(3)(4)] 5 4.3107
Using Exponential Notation in CalculationsIn calculations, you can treat the coefficient and exponents separately and apply the properties of exponents (see earlier section on logarithms).
1. To multiply exponential numbers, multiply the coefficients, add the exponents, and reconstruct the number in standard exponential notation:
solving QUAdrAtic EQUAtionsA quadratic equation is one in which the highest power of x is 2. The general form of a quadratic equation is
ax2 bx c 5 0
where a, b, and c are numbers. For given values of a, b, and c, the values of x that satisfy the equation are called solutions of the equation. We calculate x with the qua-dratic formula:
x 52b 6"b2 2 4ac
2a
We commonly require the quadratic formula when solving for some concentration in an equilibrium problem. For example, we might have an expression that is rearranged into the quadratic equation
4.3x2 0.65x 8.7 5 0 a b c
Applying the quadratic formula, with a 5 4.3, b 5 0.65, and c 5 8.7, gives
x 520.65 6"10.65 22 2 4 14.3 2 128.7 2
2 14.3 2The “plus or minus” sign () indicates that there are always two possible values for x. In this case, they are
x 5 1.3 and x 5 1.5
In any real physical system, however, only one of the values will have any meaning. For example, if x were [H3O], the negative value would give a negative concentra-tion, which has no physical meaning.
A-4 appendix a • Common Mathematical Operations in Chemistry
graphIng Data In thE ForM oF a straIght lInEVisualizing changes in variables by means of a graph is used throughout science. In many cases, it is most useful if the data can be graphed in the form of a straight line. Any equation will appear as a straight line if it has, or can be rearranged to have, the following general form:
y mx 1 b
where y is the dependent variable (typically plotted along the vertical axis), x is the independent variable (typically plotted along the horizontal axis), m is the slope of the line, and b is the intercept of the line on the y axis. The intercept is the value of y when x 0:
y m(0) 1 b b
The slope of the line is the change in y for a given change in x:
Slope (m) 5y2 2 y1
x2 2 x15Dy
Dx
The sign of the slope tells the direction of the line. If y increases as x increases, m is positive, and the line slopes upward with higher values of x; if y decreases as x increases, m is negative, and the line slopes downward with higher values of x. The magnitude of the slope indicates the steepness of the line. A line with m 3 is three times as steep (y changes three times as much for a given change in x) as a line with m 1. Consider the linear equation y 2x 1 1. A graph of this equation is shown in Figure A.1. In practice, you can find the slope by drawing a right triangle to the line, using the line as the hypotenuse. Then, one leg gives y, and the other gives x. In the figure, y 8 and x 4. At several places in the text, an equation is rearranged into the form of a straight line in order to determine information from the slope and/or the intercept. For exam-ple, in Chapter 16, we obtained the following expression:
ln 3A 403A 4t
5 kt
Based on the properties of logarithms, we have
ln [A]0 2 ln [A]t kt
Rearranging into the form of an equation for a straight line gives
ln [A]t 2kt 1 ln [A]0
y mx 1 b
Thus, a plot of ln [A]t vs. t is a straight line, from which you can see that the slope is 2k (the negative of the rate constant) and the intercept is ln [A]0 (the natural logarithm of the initial concentration of A). At many other places in the text, linear relationships occur that were not shown in graphical terms. For example, the conversion of temperature scales in Chapter 1 can also be expressed in the form of a straight line:
°F 95 °C 1 32
y mx 1 b
Figure A.1
14
12
10
8
6
4
2
y 2x 1 1
Slope 8⁄4 2
Intercept
20
2
2 4 6
x
y
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3.2 Section • Determining the Formula of an Unknown Compound 5
Standard Thermodynamic Values for Selected Substances*
(continued)*All values at 298 K, except for acetylsalicylic acid, which is at 37ºC (310 K) in 0.15 M NaCl.†Acidic (ionizable) proton(s) shown in red. Structures have lowest formal charges. Benzene rings show one resonance form.
Equilibrium Constants for Selected Substances*
Name and Formula Lewis Structure† Ka1 Ka2 Ka3
Dissociation (Ionization) Constants (Ka) of Selected Acids
Appendix C
A-8
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Appendix C • Equilibrium Constants for selected substances A-9
Equilibrium Constants for Selected Substances*
H C HO
O
H O C HO
O
C C
HH
H
O
H
C
H
H
H O C HO
O
C
H
O C HO
O
H C N
H F
HSH
ClOH
IOH
C O
H
CO
OO H
H HC C
O
C HOC
H O
H
H
H
C
H
O
OIOH
OH Br
CH O
O O
C
O
C
C
H
O
C
H
C
H
H H
HO
H
O
(continued )
Citric acid HOOCCH2C(OH)(COOH)CH2COOH
7.431024 1.731025 4.031027
Formic acid HCOOH
1.831024
Glyceric acid HOCH2CH(OH)COOH
2.931024
Glycolic acid HOCH2COOH
1.531024
Glyoxylic acid HC(O)COOH
3.5 31024
Hydrocyanic acidHCN
6.2310210
Hydrofluoric acidHF
6.831024
Hydrosulfuric acidH2S
931028 1310217
Hypobromous acidHBrO
2.331029
Hypochlorous acidHClO
2.931028
Hypoiodous acidHIO
2.3310211
Iodic acidHIO3
1.631021
Lactic acidCH3CH(OH)COOH
1.431024
Maleic acidHOOCCH CHCOOH
1.231022 4.731027
Name and Formula Lewis Structure† Ka1 Ka2 Ka3
Dissociation (Ionization) Constants (Ka) of Selected Acids
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A-10 Appendix C • Equilibrium Constants for selected substances
HOCC C
OO
OH
H
C C
C
CC
C C
H
H
H
H
HO
C C
H
H
C
O
H H
C C
C
C
C C
H
HH
HO
O H
O
PH HO
OH
O H
O
P HO
C
H
H
C
H
H
C
O
HOH
C HOC
H
H
CH
OO
C C
H
H
C
H
H
C HO
OO
OH
O
O
SH HOO
O
SH HOO
HOCC C
H
H
OO
OH
N OOH
Malonic acidHOOCCH2COOH
1.431023 2.031026
Nitrous acidHNO2
7.131024
Oxalic acidHOOCCOOH
5.631022 5.431025
PhenolC6H5OH
1.0310210
Phenylacetic acidC6H5CH2COOH
4.931025
Phosphoric acidH3PO4
7.231023 6.331028 4.2310213
Phosphorous acidHPO(OH)2
331022 1.731027
Propanoic acidCH3CH2COOH
1.331025
Pyruvic acidCH3C(O)COOH
2.831023
Succinic acidHOOCCH2CH2COOH
6.231025 2.331026
Sulfuric acidH2SO4
Very large 1.031022
Sulfurous acidH2SO3
1.431022 6.531028
(continued )
Name and Formula Lewis Structure† Ka1 Ka2 Ka3
Dissociation (Ionization) Constants (Ka) of Selected Acids (continued )
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Appendix C • Equilibrium Constants for selected substances A-11
NH
H
H
H
C C
C
C
C C
H
H
H
H
NC
H
H
NC
H
H
C
H
H
C
H
HH
C
H
H
H H
NC
H
H
C
H
HH
HH
N
H
H
H
C
H
H
HCOH
HN
H
C
H
H
C
H
H
H
N C
H
HH
NC
H
H H
HH
HNC
H
H H
H
HN
C C C
H
H
H
HC
H
H
H H
H
H
HN
H
C
H
H
C
H
H
C
H
H
H
C C
C
C
C
N HH
H
H
HH H
H H
H
H
(continued )†Blue type indicates the basic nitrogen and its lone pair.
Ammonia NH3
1.7631025
Aniline C6H5NH2
4.0310210
Diethylamine(CH3CH2)2NH
8.631024
Dimethylamine(CH3)2NH
5.931024
EthanolamineHOCH2CH2NH2
3.231025
EthylamineCH3CH2NH2
4.331024
EthylenediamineH2NCH2CH2NH2
8.531025 7.131028
Methylamine CH3NH2
4.431024
tert-Butylamine(CH3)3CNH2
4.831024
Piperidine C5H10NH
1.331023
n-PropylamineCH3CH2CH2NH2
3.531024
Name and Formula Lewis Structure† Kb1 Kb2
Dissociation (Ionization) Constants (Kb) of Selected Amine Bases
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A-12 Appendix C • Equilibrium Constants for selected substances
H
C C C
H
H H H
H
H H
H
HN
HHHH
N
N
H
C
H
C
H
C
H
H
H
C C
C
C
C C
H
H
H
H
N
NC
H
H
C C
H
H
C
H
H
C
H
H
CH H
H H
H
H
H
NC C
H
H
H
H
H
H
CH H
H
Isopropylamine(CH3)2CHNH2
4.731024
1,3-PropylenediamineH2NCH2CH2CH2NH2
3.131024 3.031026
Pyridine C5H5N
1.731029
Triethylamine(CH3CH2)3N
5.231024
Trimethylamine(CH3)3N
6.331025
Fe31
Sn21
Cr31
Al31
Cu21
Pb21
Zn21
Co21
Ni21
Fe(H2O)631(aq)
Sn(H2O)621(aq)
Cr(H2O)631(aq)
Al(H2O)631(aq)
Cu(H2O)621(aq)
Pb(H2O)621(aq)
Zn(H2O)621(aq)
Co(H2O)621(aq)
Ni(H2O)621(aq)
631023
431024
131024
131025
331028
331028
131029
2310210
1310210
Ag(CN)22
Ag(NH3)21
Ag(S2O3)232
AlF632
Al(OH)42
Be(OH)422
CdI422
Co(OH)422
Cr(OH)42
Cu(NH3)421
Fe(CN)642
Fe(CN)632
Hg(CN)422
Ni(NH3)621
Pb(OH)32
Sn(OH)32
Zn(CN)422
Zn(NH3)421
Zn(OH)422
3.031020
1.73107
4.731013
4 31019
3 31033
4 31018
1 3106
5 3109
8.031029
5.631011
3 31035
4.031043
9.331038
2.03108
8 31013
3 31025
4.231019
7.83108
3 31015
Name and Formula Lewis Structure† Kb1 Kb2
Dissociation (Ionization) Constants (Kb) of Selected Amine Bases (continued )
Free Ion Hydrated Ion Ka
Dissociation (Ionization) Constants (Ka) of Some Hydrated Metal Ions
Complex Ion Kf
Formation Constants (Kf) of Some Complex Ions
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Appendix C • Equilibrium Constants for selected substances A-13
*All values at 298 K. Written as reductions; E value refers to all components in their standard states: 1 M for dissolved species; 1 atm pressure for the gas behaving ideally; the pure substance for solids and liquids.
BA
BABA
BA
BA
BABA
BABA
BA
BA
BABA
BABA
BA
BABABABA
BABA
BABABABABABABA
BA
BABABABA
BA
BABA
BABA
BABA
BABABA
BABA
BABA
Standard Electrode (Half-Cell) Potentials*Half-Reaction E 8 (V)
Appendix D
A-14
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