Question

Taking S as entropy, T as temperature, P as pressure, and V as volume, match Column I with Column II.
Column I
(A) Gibbs Free Energy
(B) Helmholtz Free Energy
(C) Enthalpy
(D) Internal Energy
Column II
(1) depends on T, V and composition
(2) depends on T, P and composition
(3) depends on S, P and composition
(4) depends on S, V and composition

• A. A – 2, B – 1, C – 3, D – 4
• B. A – 4, B – 3, C – 2, D – 1
• C. A – 3, B – 1, C – 4, D – 2
• D. A – 2, B – 1, C – 4, D – 3

Hint:

Use a mnemonic like the "Thermodynamic Square" to remember these relationships. Another simple way is to start from \(dU = TdS - PdV\) and remember the definitions: \(H = U+PV\), \(A = U-TS\), \(G = H-TS\). You can then quickly derive the differential forms and identify the natural variables.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks to match thermodynamic potentials (Internal Energy, Enthalpy, Helmholtz Free Energy, Gibbs Free Energy) with their natural variables. The natural variables are the set of variables that allow the potential to be expressed in its most simple and fundamental form, derived from the first and second laws of thermodynamics.
Step 2: Key Formula or Approach:
The fundamental thermodynamic relation for internal energy (U) is \(dU = TdS - PdV\). This shows that the natural variables for U are S and V. From this, we can derive the other potentials and their natural variables through Legendre transformations.
- Internal Energy: \(U = U(S, V)\)
- Enthalpy: \(H = U + PV\). Its differential is \(dH = TdS + VdP\). Natural variables are \(S\) and \(P\).
- Helmholtz Free Energy: \(A = U - TS\). Its differential is \(dA = -SdT - PdV\). Natural variables are \(T\) and \(V\).
- Gibbs Free Energy: \(G = H - TS = U + PV - TS\). Its differential is \(dG = -SdT + VdP\). Natural variables are \(T\) and \(P\).
For multicomponent systems, a composition term (\(\sum \mu_i dN_i\)) is added to each differential, so each potential also depends on composition.
Step 3: Detailed Matching:
- (A) Gibbs Free Energy (G): Its natural variables are Temperature (T) and Pressure (P). Thus, G depends on T, P, and composition. A matches with 2.
- (B) Helmholtz Free Energy (A or F): Its natural variables are Temperature (T) and Volume (V). Thus, A depends on T, V, and composition. B matches with 1.
- (C) Enthalpy (H): Its natural variables are Entropy (S) and Pressure (P). Thus, H depends on S, P, and composition. C matches with 3.
- (D) Internal Energy (U or E): Its natural variables are Entropy (S) and Volume (V). Thus, U depends on S, V, and composition. D matches with 4.
The correct matching is: A-2, B-1, C-3, D-4. This corresponds to option (A).
Step 4: Why This is Correct:
The matching is based on the fundamental definitions and differential forms of the thermodynamic potentials. Option (A) correctly pairs each potential with its set of natural variables.