MATHEMATICAL PROBLEM PAGE Directed by EDWARD L. HAENISCH Montana State College, Bozeman, Montana
T
HE RESULT of .an experimental measurement should be expressed to the proper number of significant figures, i. e., to such a number of figures that all except the last are known with certainty. Thus if the weight of an object is given as 1.21 grams the weighing was made only to the closest 10 mg., while 1.2144 would indicate a weighing made to the closest 0.1 mg. When several factors are combined the result must contain the proper number of figures. Convenient rules are: I. In addition or subtraction extend the significant figures in each term and in the s u p or difference only to the point corresponding to that uncertain figure occumng farthest to the left relative to the decimal point. Thus : 203.1
by getting a consistent set of results or, in other words, obtaining checks. Probability considerations show that the arithmetical mean serves as the best average value of a set of distinct results. The precision or consistency of the values can be judged by a number of means. The most useful is the "average deviation." This is defined as follows, where d is the difference between the average and the individual result regardless of sign and n is the number of results. o = average deviation of
a single result =
+
Zd . n
The "mean square error," m, is the error whose square is the average of the squares of all the errors.
+ 7.21 + 0.3134 = 210.6
m = mean square error of a single result =
*
-\l-
n-1
11. Retain in a product or a quotient as many significant figures as appear in the least accurate factor. (The % precision of the product or quotient cannot be greater than % precision of least accurate factor.) Thus : 1.31 X 20.315 = 26.6
The "probable error," r, is the error such that the number of errors greater than r is equal to the number of errors less than r. It i s not the error most likely to
In any quantitative experiment both determinate and indeterminate errors are present. Determinate or constant errors arise from errors inherent in the procedure, instrumental errors, etc.; they can be eliminated only by changing methods, calibration of instruments, or running blanks and controls. Indeterminate or accidental errors are due to causes over which the experimenter has no control. They can be eliminated
Likewise we can compute the average deviation, the mean square error, or the probable error of the arithmetical mean. These are obtained by dividing the appropriate quantity for a single observation by the square root of the number of observations. For example:
OCGUT.
r = probable error of a single result = += 0.6745
A
=
average deviation of the arithmetical mean =
11-1
+
Zd 4%'
n
Sometimes a result deviates quite widely from the rest. Fales quotes a convenient rule. Omit the doubtful determination and compute the arithmetical mean and the average deviation of a single result. If the difference between the arithmetical mean and the doubtful result is more than four times the average deviation discard the result. (The above cannot be a p w to less than four results.) BIBLIOGRAPHY
HAMILTON AND SIMPSON, "Calculations of quantitative analysis," McGraw-Hill Book Co., New York City, 1927,Chap. I. FALES. "Inorganic quantitative analysis," Century Co., New York City, 1925, Chap. IV. FLNDLAY, "Practical physical chemistry," Longmans, Green & Co., London, 1933,Chap. I. MELLOR, "Higher mathematics for students of chemistry and physics'" Longmans' Green & CO.' London' 1922, Chap. IX. for physical chemistry," McGraw-Hill Book Co., New York City, 1928, Chap. XX. (The last two references will hereafter be referred to as "MELLOR"and "DANIELS.") PROBLEMS
1. Calculate the following. Express the answers. with the correct number of significant figures. (a)
1211.1
+ 0.7641 + 1.12
(b) 7.516 X 1.31 X 0.2954 (c) (1.276
x
0.00056) - (1.2
x
10-O)
- (0.0023456 X
o ,0128)
2. The temperature of a thermostat was read at two-minute intervals for a oeriod of 20 minutes. T
i
TImpnnlv,a ( B c r k m o n R a o ~ i n p ~
Tsmpnolrrr
3.181 3.168 3.159 3.180 3.152 3.102
3
2
8
3.152 3.158 3.162 3.181 3.155
7 8 9 10
Calculate the average temperature, the average deviation of a single temperature, the average deviation of the arithmetical mean, the mean and the probable errors of a single result, and of the arithmetical mean. 3, The sought in quantitative is, in general, 1/1000. A volume measured from a buret can usually be precise to within about +0.02 cc. Similarly, weighings can be made to *0.2 mg. A sample of iron ore contains about 10% iron. The iron is to be determined by titration with potassium dichromate which is 0,1000 N, How large a sample of ore must be taken for analysis to attain the desired precision? 4. An analyst obtains the following values for the normality of an HC1 solution when he standardizes it against N&CO3: 0.1041; 0.1043; 0.1044; 0.1040; 0.1032. What is the normality of the acid? Should any of the results be discarded? 5. An empirical equation which states the frequency of occurrence,y, of an error, x, is: = ke-m**
where k and h are constants. Plot a set of curves and determine the effect of varying k and h. (Values for ed"" may be obtained from tables or calculated by the use of logarithms.) Show why h should be termed the .. "modulus of precision."
SOLUTIONS TO JANUARY PROBLEMS
( +$)(V"r-b)=RT
1. (a) Van der Waal's Equation is P
(dynes em.?) (cm.9 cm.' x cm.
=
V , denotes the molar volume under the conditions of the experiment.
(e) r =
7
M u )
(T,-T-6)
a must have same units as P. Vma If P is in atmospheres,
dyne wn.-I (cm.5 mol-1)Vr deg.
= dyne cm. mol
a cm.=mol-a atm. mol-'atm. for - = = atm. V', (em.' mol-I)%
V = cm.aequiv.-l
S = ohm-'cm.-l.
A = VS = cm.' equiv.-' ohm-> cm.?
= erg mol-'/a
2.
-% deg.?
deg.-'
1 atm. = 2116.21h.ft.-' 22.4 liters = 0.7912 cu. ft. = (ft.)' 2116.2 1b.ft.-2 X 0.7912 (ft.)s = 5.993 ft. Ib. mol-' 1 mol X 273 deg. deg. -I
R
=
P
dx
= ahm-lm.1
equiv. -I.
. P has. JP P- ISunitless.
dynes cm.? (g. mol-1cm.r g.-1)'/. deg.
= -
Vm in ~m.~rnol-1, o must he cm.6
(6) Equivalent Conductivity is defined as A = VS,where V is the volume required to contain one equivalent of electrolyte.
=
sec. = dynes cm.? see
ZR T
= dynes-.-2
ergs
g. mol-I mol-' deg.-'
deg.
= g. em. se~.-~cm.-~
= g. an.-%ec.-'
The equation is correct. (b) V = cm. see.? Aa = ohm-'cm.' equiv.?
E
= volt cm.-1
F = coulombs equiv.? (From Ohm's Law volt ohm-' = amp.)
-
A& cm.3) olt - - (~hm-~cm.~equiv.-~)(v
F
coulombs equiv.?
ohm-') cm. - (voltamp. = cm. set.-I sec. The equation is correct. In K.,. 4, d In K.,. has no units. Therefore, the units of ddT
are
deg. -I
This may he obtained on the right-hand side by taking AH-
RT
-
cal. molP cal. m o l F deg.? deg."
deg'-''
