DIFFERENCE QUOTIENTS AND THEIR APPLICATION TO CERTAIN TYPES OF EMPIRICAL EQUATIONS IN PHYSICAL CHEMISTRY
The interpolation and representation of functions by algebraic polynomials as based on the theory of finite differencesis well known, but this method fails for unequal spacing of the variables. The method of difference quotients discussed by Ballantine' is a generalization which will apply to unequally spaced intervals, or even to cases where the value of the first derivative of the function is known for one or more points. In this paper is shown the application of this more general method to equations such as are often used in the tabulation of scientific data, namely, the equation ~ = a + b x + c x ~ + ....,~ d l o g P = A / T . + ~ ~ + c .
If we let x and y represent the independent and the dependent variable, respectively, we must calculate the following finite differencesand difference quotients. (We refer the reader to the original paper for mathematical proof.) a - 2,. A,z, = %a--21; A2%, = x2
1
in general A%
=
- Z" 7, also A,y, = y,
Z"+l
- ys; and
,
in general, for first order differences only, To calculate the second order differences of y , we need first order difference (Distinction from the method of finite quotients, Qly. = A,y,JA,x,. differences.) Second order differences are then Aty,, = Qly, + - Qly,. For third order differences, we repeat the process, using second order difference quotients. In general, Ah + 1 y , = Q o , + I - Qhy, and Q,y, = These differences and difference quotients can be. arranged A&/Ahx.. in the following manner:
The equation connecting x and y will be given by
(2
- X")
' BALLANTINE, Am. Math. Monthly, 2 4 5 3 (1919). 2094
VOL.9, No. 12
DIFFERENCE QUOTIENTS
2095
As an example, let us assume that y has the values 7, 5, 35, and 47, for x equal t o 0; 2,7, and 8. We will then have
+
and the equation:will be y = .7- 1(x-0) 2/2! (x-0)(3~-2), which 7. simplifies t o y = x2 - 3x The method exemplified with integer numbers is applicable to such cases as finding an equation for the thermocouple corrections tabulated in Table I.
+
TABLE I Thermocouple Corrections Oblnvrd h4imouoP1
TDA.
273.1 209.06 151.43 84.60
Minaollr Tabla
f r m
...
...
4469 7734 10523
4497 7798 10623
Carccrion
0 28 64 100
For this example, the differences and difference quotients are arranged as follows (e' = e X e'
my.
We obtain the equation mv. = 0
+ 6.2656' + 0.6156e' (e' - 4.469) - 0.0289e' (e' - 4.469)(s' - 7.734)
which can also be written as mv. = 2.52(e/1000)
+ 0.968(~/1000)~- 0.0289(e/1000)"
The method of difference quotients can also he applied to those equations which with some slight modification or change of variable can be written in the form just exemplified. For example, the coefficientsin the equation, log P = A / T BT C, can be determined by this method if we consider T log P as the dependent variable. For the sake of an example, the vapor pressure of liquid oxygen is given in Table 11, and applying the method of difference quotients, we obtain the equation
+
+
log P,",: = -417 925 - 0.006332 T
T
(Concluded on page 2098)
+ 8.0875
2098
JOURNAL OF CHEMICAL EDUCATION
DECEMBER, 1932
(Continued from @ge 2095)
TABLE Il Vapor Pressure of Oxygen2 T'A.
T log F
Pnn-
71.72 81.09 90.15
64.01 263.19 760. W
129.545 196.260 259.710
The method of difference quotients can also be used in those cases where the value of the first derivative is known. In theory, this amounts to giving the value for the difference quotient for two values of y which are identical. This is evident because d$/dx is the limiting value of (y2-y,) / ( X P - x , ) ,as the two points (x,,y l ) and (22, fl) approach each other. Assume, for example, that when x has the value 1 and 4, y has the value 3 at both points. Also that the value of the first derivative is -3 for x = 1. The calculation and result are as follows : x 1
Y 3
1
3
4
3
0
3/2 y = 3 - 3(x - 1)
y=x2-5~+7
0 0
-3 0
3
2
+ 2/2!(x - l ) ?
Such cases are not likely to arise for the fitting of curves to data, but may be of interest mathematically. Ballantine (loc. cit.) has shown how to interpolate a logarithmic function more quickly, since in this case the first derivative can be calculated readily, namely, if y = log x, then dy/dz = 0.43129/x.
The method of difTerence quotients is applicable only to those functions which can be expressed as a polynomial in x. "Int. Crit. Tables," Val. 3, p. 203.
