The fundamental thermodynamic relation for a rubber band is given by 𝒅𝑼 = 𝑻𝒅𝑺 + 𝝉𝒅𝑳, where 𝑻 is the absolute temperature, 𝑺 is the entropy, 𝝉 is the tension in the rubber band, and 𝑳 is the length of the rubber band. Which one of the following relations is CORRECT:

The correct relation is:
${\left(\frac{\partial T}{\partial L}\right)}_S={\left(\frac{\partial \tau }{\partial S}\right)}_L$

Step-by-Step Derivation:

1. Understand the fundamental thermodynamic relation:
We are given the fundamental thermodynamic relation for a rubber band as:
$dU=TdS+\tau dL$
where:
• U is the internal energy,
• T is the absolute temperature,
• S is the entropy,
• τ is the tension in the rubber band, and
• L is the length of the rubber band.

2. Apply the condition for an exact differential:
Since the internal energy U is a thermodynamic state function, its differential dU is an exact differential.
In general, if a differential expression is written in the form:
$dZ=Mdx+Ndy$
where dZ is an exact differential, then the coefficients M and N satisfy the Euler reciprocity relation (or Maxwell relation criterion):
${\left(\frac{\partial M}{\partial y}\right)}_x={\left(\frac{\partial N}{\partial x}\right)}_y$

3. Compare terms to find the variables:
Comparing the rubber band relation dU = T dS + τ dL with the exact differential form dZ = M dx + N dy, we map:
• Z = U
• M = T and x = S
• N = τ and y = L

4. Substitute the variables to obtain the relation:
Substituting these variables into the reciprocity relation yields:
${\left(\frac{\partial T}{\partial L}\right)}_S={\left(\frac{\partial \tau }{\partial S}\right)}_L$
This matches the correct relation.