ABSTRACT

In Chapter 2, we obtained the thermodynamic equations in terms of dQ which give the amount of heat absorbed by a system when the independent variables change. However, the formulas involve derivatives of the internal energy, which is not directly measurable. We can obtain more practical results by rewriting the equations by exploiting the fact that dS = dQ/T is an exact differential. We illustrate the method using (2.6), with T and V as independent variables:

dQ = T dS = CV dT + [(

∂U ∂V

) T + P ]

dV . (3.1)

Dividing both sides by T , we obtain an exact differential:

dS = CV T

dT + 1 T

[( ∂U ∂V

) T + P ]

dV , (3.2)

which must be of the form

dS = ∂S ∂T

dT + ∂S ∂V

dV . (3.3)

Here we have suppressed the subscripts on partial derivatives. Thus, we can identify ∂S ∂T

= CV T

,

∂S ∂V

= 1 T

[ ∂U ∂V

+ P ]

. (3.4)

Since differentiation is a commutative operation, we have ∂

∂V ∂S ∂T

= ∂ ∂T

∂S ∂V

. (3.5)

Hence ∂

∂V CV T

= ∂ ∂T

[ 1 T

( ∂U ∂V

+ P )]

. (3.6)

be V since T is kept differentiation. Using CV = ∂U/∂T , we can rewrite the last equation in the form

1 T

∂V ∂U ∂T

= − 1 T 2

[ ∂U ∂V

+ P ] + 1

T

[ ∂

∂V ∂U ∂T

+ ∂P ∂T

] . (3.7)

After canceling identical terms on both sides, we obtain the energy equation( ∂U ∂V

) T = T

( ∂P ∂T

) V − P, (3.8)

where we have restored the subscripts. The derivative of the internal energy is now expressed in terms of readily measurable quantities.