Understanding the Helmholtz Function in Thermodynamics

The Helmholtz function, often denoted by \(A\) or \(F\), is a thermodynamic potential that is particularly useful when dealing with systems held at a constant temperature and volume. It provides a measure of the useful work obtainable from a closed thermodynamic system at a constant temperature and volume.

The Helmholtz function is defined in terms of other thermodynamic properties: internal energy, temperature, and entropy.

Defining the Helmholtz Function (A)

The standard definition of the Helmholtz function is given by the equation:

\(A = U - TS\)

Where:

• \(A\) is the Helmholtz function (sometimes denoted as \(F\)).
• \(U\) is the internal energy of the system.
• \(T\) is the absolute temperature of the system.
• \(S\) is the entropy of the system.

This equation shows that the Helmholtz function is obtained by subtracting the product of temperature and entropy (TS) from the internal energy (U).

Comparing with the Given Options

Let's look at the provided options and compare them with the definition of the Helmholtz function \(A = U - TS\):

1. u – Ts: This option uses lowercase letters (\(u\) for internal energy, \(s\) for entropy, \(T\) for temperature), which are commonly used for specific properties (per unit mass). However, the form matches the definition: specific internal energy minus temperature times specific entropy. This corresponds directly to the definition \(A = U - TS\) using specific properties.
2. – sdT + vdp: This expression looks like a differential form, possibly related to one of the thermodynamic potentials, but it does not match the standard definition of the Helmholtz function itself. The differential form of the Helmholtz function is \(dA = -SdT - PdV\) (or \(da = -sdT - pdv\) for specific properties). This option does not match either the standard form or its differential.
3. h – Ts: This option uses \(h\), which typically represents enthalpy (\(H = U + PV\)). The expression \(H - TS\) is the definition of the Gibbs free energy (G), not the Helmholtz function.
4. u + pv: This option represents the definition of enthalpy (\(H = U + PV\)), using specific properties (\(u\) for specific internal energy, \(p\) for pressure, \(v\) for specific volume). This is enthalpy, not the Helmholtz function.

Conclusion

Based on the definitions of thermodynamic potentials, the expression for the Helmholtz function is \(A = U - TS\) or \(F = U - TS\). Using lowercase letters for specific properties, this becomes \(a = u - Ts\).

Comparing the options, the expression that matches the definition of the Helmholtz function (using specific properties) is \(u – Ts\).