ABSTRACT
Thermodynamics becomes a mathematical science when we regard the state functions, such as the internal energy, as continuous differentiable functions of the variables P, V and T . The constraint imposed by the equation of state reduces the number of independent variables to two. We may consider the internal energy to be a function of any two of the variables. Under infinitesimal increments of the variables, we can write
dU(P, V ) = ( ∂U ∂P
) V
dP + ( ∂U ∂V
) P
dV ,
dU(P, T ) = ( ∂U ∂P
) T
dP + ( ∂U ∂T
) P
dT ,
dU(V , T ) = ( ∂U ∂V
) T
dV + ( ∂U ∂T
) V
dT ,
(2.1)
where a subscript on a partial derivative denotes the variable being held fixed. For example, (∂U/∂P)V is the derivative with respect to P at constant V . These partial derivatives are thermodynamic coefficients to be taken from experiments.
