Modern Thermodynamics

- Chapter 1

42

Appendix 1.1: Partial Derivatives

Derivatives of many variables

When a variable such as energy U(T,V, Nk) is a function of many variables V, T and Nk, its

partial derivative with respect to each variables is defined by holding all other variables constant. Thus,

for example, if

U(T,V,N)

=

5

2

NRT − a

N2

V

then the partial derivatives are:

∂U

∂T

V,N

=

5

2

NR

(A1.1.1)

∂U

∂NV,T

=

5

2

RT − a

2N

V

(A1.1.2)

∂U

∂VN,T

= a

N2

V2

(A1.1.3)

The subscripts indicate the variables that are held constant during the differentiation. In cases where the

variables being held constant are understood, the subscripts are often dropped. The change in U, i.e. the

differential dU, due to changes in N, V and T is given by:

dU =

∂U

∂T

V,N

dT +

∂U

∂VT,N

dV +

∂U

∂NV,T

dN

(A1.1.4)

For functions of many variables, there is a second derivative corresponding to every pair of variables:

∂2U

∂T∂V

,

∂2U

∂N∂V

,

∂2U

∂T2

.. etc. For the "cross derivatives" such as

∂2U

∂T∂V

that are derivatives with respect

to two different variables, the order of differentiation does not matter. That is

∂2U

∂T∂V

=

∂2U

∂V∂T

(A1.1.5)