Entropy (classical Thermodynamics) - Entropy Measurement

Entropy Measurement

For simplicity, we examine a uniform closed system, whose thermodynamic state is determined by its temperature T and pressure P. A change in entropy can be written as

The first contribution depends on the heat capacity at constant pressure C_P through

This is the result of the definition of the heat capacity by δQ = C_PdT and TdS = δQ. For rewriting the second term we use one of the Maxwell relations

and the definition of the volumetric thermal-expansion coefficient

so that

With this expression the entropy S at arbitrary P and T can be related to the entropy S₀ at some reference state at P₀ and T₀ according to

In classical thermodynamics the entropy of the reference state can be put equal to zero at any convenient temperature and pressure. E.g., for pure substances, one can take the entropy of the solid at the melting point at 1 bar equal to zero. From a more fundamental point of view, the third law of thermodynamics suggests that there is a preference to take S = 0 at T = 0 (absolute zero) for perfectly ordered materials such as crystals.

In order to determine S(P,T) we followed a specific path in the P-T diagram: first we integrated over T at constant pressure P₀, so that dP=0, and in the second integral we integrated over P at constant temperature T, so that dT=0. As the entropy is a function of state the result is independent of the path.

The above relation shows that the determination of the entropy requires knowledge of the heat capacity and the equation of state (which is the relation between P,V, and T of the substance involved). Normally these are complicated functions and numerical integration is needed. In simple cases it is possible to get analytical expressions for the entropy. E.g., in the case of an ideal gas, the heat capacity is constant and the ideal-gas law PV = nRT gives that α_VV = V/T = nR/p, with n the number of moles and R the molar ideal-gas constant. So, the molar entropy of an ideal gas is given by

In this expression C_P now is the molar heat capacity.

The entropy of inhomogeneous systems is the sum of the entropies of the various subsystems. The laws of thermodynamics hold rigorously for inhomogeneous systems even though they may be far from internal equilibrium. The only condition is that the thermodynamic parameters of the composing subsystems are (reasonably) well-defined.