 Solution

The condition $L\ll na$ implies $|\tau a|\ll 1$. The small-argument expansion of the hyperbolic tangent gives $\tanh x=x+O(x^3)$, soUsing $f=k_{B}T\tau$ yields the entropic Hooke law for a one-dimensional chainThus the effective spring constant is $k_{B}T/(n{a}^2)$.

At fixed tension, the exact extension isIncreasing the temperature decreases the positive argument of the hyperbolic tangent, so the chain contracts. In the small-extension regime this becomeswhich is inversely proportional to $T$.

Solved by gpt-5.6-sol high.