The Chemistry Maths Book, Second Edition

40 Chapter 2Algebraic functions

EXAMPLE 2.12The van der Waals equation

The equation of state for a ‘slightly imperfect gas’ is

(2.8)

In this case, both Tand pare easily expressed as explicit functions of the other variables:

For V, equation (2.8) can be rearranged into

which is a cubic equation in V. It is possible to write down explicit solutions of a

cubic equation, but these are complicated and seldom used. In this case, it is most

convenient to regard equation (2.8) as defining Vas an implicit function of p, T, and

n. For any set of values of the independent variables and of the constants, equation

(2.8) can be solved numerically by an iterative method such as the Newton–Raphson

method described in Chapter 20.

0 Exercises 32–35

2.5 Polynomials

The general polynomial of degreenhas the form

f(x) 1 = 1 a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-1+ 1 a

n

x

n

(2.9)

where the coefficientsa

0

, 1 a

1

, 1 =, 1 a

n

are constants, and nis a positive integer. If

n 1 = 10 the function is the constant a

0

. The polynomial is often written in short-hand

notation as

(2.10)

where the symbol ∑represents summation. The notation tells us to add together the

termsa

i

x

i

in which the integer variable itakes in turn the values0, 1, 2,1=1,n:

=a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-1+ 1 a

n

x

n

(remembering thatx

0

1 = 11 andx

1

1 = 1 x).

i

n

i

i

n

n

ax ax ax ax a x

=

∑

=++++

0

0

0

1

1

2

2

()()() ()

fx ax

i

n

i

i

()=

=

∑

0

Vnb

RT

p

V

na

p

V

nab

p

32

23

−+ 0













+−=

T

nR

p

na

V

Vnb p

nRT

Vnb

na

V

=+

















−=

−

−

1

2

2

2

2

(),

p

na

V

• Vnb nRT

















−− =

2

2

() 0