Page 148 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 6 DIFFERENTIATION 125

X
Y=
(1 + x)&
exp[(x - p)2 / 202]
Y= 04%

7. Show that the slope of the logistic equation

y=- 1
1 + e-"
at its midpoint (the Hall slope) is equal to a/4.

8. The van der Waals equation is an equation of state that applies to real gases.
For 1 mole of a gas, the van der Waals equation is
( vr)
(V-b)=RT
P+-

where R is the gas constant (0.0821 L atm K-' mol-') and T is the Kelvin
temperature. The constants a and b are constants particular to a given gas,
and correct for the attractive forces between gas molecules, and for the
volume occupied by the gas molecules, respectively. For methane (CK), the
constants are a = 2.253 L2 atm and b = 4.278 x L. Using the rearranged
form of the van der Waals equation

calculate the pressure of 1 mole of methane as a function of container volume
at 0°C (273 K) at suitable volumes from 22.4 L to 0.05 L. Use one of the
custom functions described in this chapter to calculate the first and second
derivatives of the P-V relationship. Compare with the exact expressions
2a
dP - RT +-
dV (V-b)2 V3

d2P 2RT 6a
-- - --
dV2 (V-b)3 V4