Thermodynamic Relation for a Rubber Band

The question provides the fundamental thermodynamic relation for a rubber band:

\[dU = TdS + \tau dL\]

In this equation, each term represents a specific physical quantity:

• \(U\) represents the internal energy of the rubber band, which is a state function.
• \(T\) is the absolute temperature, an intensive property.
• \(S\) is the entropy, an extensive property.
• \(\tau\) is the tension in the rubber band, an intensive property.
• \(L\) is the length of the rubber band, an extensive property.

This equation is a fundamental differential form in thermodynamics. It describes how the internal energy of the rubber band changes when its entropy and length are varied. The term \(TdS\) represents the heat transfer, and \(\tau dL\) represents the work done on the rubber band when its length is changed against the tension.

Understanding Exact Differentials and Maxwell's Relations

Since \(U\) (internal energy) is a state function, its differential \(dU\) is an exact differential. For any exact differential of the form \(df = M(x, y)dx + N(x, y)dy\), there is a fundamental mathematical property:

\[\left(\dfrac{\partial M}{\partial y}\right)_x = \left(\dfrac{\partial N}{\partial x}\right)_y\]

This property is crucial in thermodynamics for deriving Maxwell's relations. Maxwell's relations are a set of equations that relate the partial derivatives of thermodynamic properties. They are derived from the fact that the second derivatives of thermodynamic potentials (like internal energy \(U\), enthalpy \(H\), Helmholtz free energy \(A\), and Gibbs free energy \(G\)) are independent of the order of differentiation.

Deriving the Correct Relation for the Rubber Band

Let's apply the exact differential condition to the given fundamental thermodynamic relation for the rubber band:

\[dU = TdS + \tau dL\]

We can compare this equation to the general form of an exact differential \(dU = \left(\dfrac{\partial U}{\partial S}\right)_L dS + \left(\dfrac{\partial U}{\partial L}\right)_S dL\). From this comparison, we can identify the coefficients:

• The coefficient of \(dS\) is \(T\). So, \(M = T\). In terms of partial derivatives, \(T = \left(\dfrac{\partial U}{\partial S}\right)_L\).
• The coefficient of \(dL\) is \(\tau\). So, \(N = \tau\). In terms of partial derivatives, \(\tau = \left(\dfrac{\partial U}{\partial L}\right)_S\).

Now, we apply the exact differential condition: \(\left(\dfrac{\partial M}{\partial y}\right)_x = \left(\dfrac{\partial N}{\partial x}\right)_y\).

In our equation, we consider \(M = T\), \(N = \tau\), \(x = S\) (corresponding to \(dx\)), and \(y = L\) (corresponding to \(dy\)).

Substituting these into the exact differential condition, we get:

\[\left(\dfrac{\partial T}{\partial L}\right)_S = \left(\dfrac{\partial \tau}{\partial S}\right)_L\]

This is a Maxwell's relation specifically derived for a system like a rubber band, where internal energy is a function of entropy and length.

Comparing with the Given Options

Let's evaluate each option based on our derived Maxwell's relation and the direct identification from the fundamental equation:

1. \(\left(\dfrac{\partial T}{\partial L}\right)_S = \left(\dfrac{\partial \tau}{\partial S}\right)_L\)
   This relation exactly matches the Maxwell's relation we derived from the fundamental thermodynamic relation \(dU = TdS + \tau dL\).
2. \(\tau = \left(\dfrac{\partial U}{\partial S}\right)_L\)
   From our initial identification, we found that \(T = \left(\dfrac{\partial U}{\partial S}\right)_L\). Therefore, this option is incorrect.
3. \(T=\left(\dfrac{\partial U}{\partial S}\right)_\tau\)
   While \(T\) is indeed \(\left(\dfrac{\partial U}{\partial S}\right)\), the subscript (constant variable) must be \(L\) (length), not \(\tau\) (tension). Thus, this option is incorrect.
4. \(\left(\dfrac{\partial T}{\partial S}\right)_L = \left( \dfrac{\partial \tau}{\partial L}\right)_S\)
   This relation does not correspond to the Maxwell's relation derived from \(dU = TdS + \tau dL\). The partial derivatives are not in the correct form for the exact differential property. Therefore, this option is incorrect.

Based on the detailed derivation and comparison, the first option is the correct relation.