PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 83

3.4 Combined formulations of the first and second laws of thermodynamics

In this section we will assume (if not otherwise stated) that every system referred to is closed
and homogeneous, that it exchanges only volume work with its surroundings, and that all the
occurring processes are reversible. For the sake of simplicity, we will not explicitly indicate
these assumptions.

3.4.1 Gibbs equations.

By combining equations (3.4) and (3.6), and the definitions (3.9), (3.12) and (3.14) of the ther-
modynamic functionsH, F, Gwe obtain relations which are called the combined formulations
of the first and second laws of thermodynamics, or the Gibbs equations:

dU = TdS−pdV , (3.33)

dH = TdS+Vdp , (3.34)

dF = −SdT−pdV , (3.35)

dG = −SdT+Vdp. (3.36)

In equation (3.33), internal energy is a function of variablesSandV. VariablesSandV will
be called thenatural variablesof functionU. The natural variables of enthalpy areSandp,
those of the Helmholtz energy areTandV, and those of the Gibbs energy areT andp.

Note:Extension of the Gibbs equations to open systems is dealt with in section6.4.1.

3.4.2 Derivatives ofU,H,F, andGwith respect to natural variables

If we consider the internal energyUas a function ofSandV, its total differential according
to (3.25) equals

dU=

(

∂U

∂S

)

V

dS+

(

∂U

∂V

)

S

dV. (3.37)