Question

Which of following is correct form of first TdS equation ?

• A. \( C_v dT + T \left(\frac{\partial V}{\partial T}\right)_V dV \)
• B. \( C_v dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \)
• C. \( C_v dT - T \left(\frac{\partial P}{\partial T}\right)_V dV \)
• D. \( C_p dT + T \left(\frac{\partial V}{\partial T}\right)_P dV \)

Hint:

There are two main TdS equations. Memorizing them is useful:
{1st TdS Equation (S in terms of T, V):} \( TdS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \)
{2nd TdS Equation (S in terms of T, P):} \( TdS = C_P dT - T \left(\frac{\partial V}{\partial T}\right)_P dP \)
Remembering that the first involves \(C_V\) and \(dV\), while the second involves \(C_P\) and \(dP\), helps to distinguish them.

Answer 1

The Correct Option is B

Step 1: Conceptual Foundation:
The TdS equations establish a relationship between entropy change (S) and other thermodynamic variables like temperature (T), volume (V), and pressure (P). They are derived from the first law of thermodynamics and Maxwell's relations. The "first" TdS equation specifically expresses dS in terms of dT and dV.

Step 2: Core Methodology:
We initiate the process by treating entropy S as a function of temperature T and volume V, denoted as \( S(T, V) \). The total differential is expressed as:
\[ dS = \left(\frac{\partial S}{\partial T}\right)_V dT + \left(\frac{\partial S}{\partial V}\right)_T dV \]
The objective is to determine the values of the two partial derivatives.

Step 3: Detailed Derivation:
1. Calculation of \( (\partial S / \partial T)_V \):
The definition of heat capacity at constant volume is \( C_V = T \left(\frac{\partial S}{\partial T}\right)_V \).
Rearranging yields: \( \left(\frac{\partial S}{\partial T}\right)_V = \frac{C_V}{T} \).

2. Calculation of \( (\partial S / \partial V)_T \):
This term is evaluated using a Maxwell relation. The Maxwell relation derived from the Helmholtz free energy (\(F = U - TS\)) is:
\[ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V \]

3. Integration into the dS equation:
The derived expressions for the partial derivatives are substituted back into the total differential equation for dS:
\[ dS = \left(\frac{C_V}{T}\right) dT + \left(\frac{\partial P}{\partial T}\right)_V dV \]

4. Derivation of the TdS form:
Multiplying the entire equation by T yields the first TdS equation:
\[ TdS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \]

Step 4: Conclusion:
The first TdS equation is correctly stated as \( TdS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \). This aligns with the expression \( C_v dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \).