The Chemistry Maths Book, Second Edition

5.8 Pressure–volume work 159

The total work done on the fluid in the cyclea 1 → 1 b 1 → 1 ais therefore

W

aba

1 = 1 W

ab

1 + 1 W

ba

1 = 12 ∆p(V

a

1 − 1 V

b

) 1 > 10 (5.68)

and the process is said to be irreversible. This is analogous to the case in ordinary

dynamics when nonconservative dissipative forces are present. The process can be

made reversible only by letting the excess pressure, ∆pin our example, approach zero.

In this (ideal) limit the work is reversibleand

(5.69)

Expansion of a gas

The equation of state of a gas is a relation of the formf(p, 1 V, 1 T) 1 = 10 amongst the

three thermodynamic quantities p, V, and T(for a given amount of gas). For example,

the equation of state for the ideal gas can be written as pV 1 − 1 nRT 1 = 10. There is

therefore some freedom in the choice of conditions under which the expansion of the

gas can occur.

isobaric expansion

Expansion of a gas can occur at constant pressure; for example, the gas can be heated

to expand against a constant external pressure such as atmospheric pressure. The

reversible work done by the gas against the external pressure is then

(5.70)

For the ideal gas,pV 1 = 1 nRTand

W 1 = 1 nR(T

b

1 − 1 T

a

) (5.71)

isothermal expansion

Expansion of a gas can occur at constant temperature; for example, if the expansion

occurs with the container immersed in a heat bath at a given temperature. To

calculate the work it is necessary to know the equation of state. For the ideal gas,

p 1 = 1 nRT 2 Vand the (reversible) work is

(5.72)

0 Exercise 56

W p dV nRT

dV

V

nRT

V

V

nRT

p

p

a

b

a

b

b

a

b

a

==ZZ= =−ln ln

WpdVpdVpVV

a

b

a

b

ba

===−ZZ()

WW pdV

ab ba

a

b

=− =−Z