Question

Rubber bands are entropic springs. Experiments show that the retractive force \(f\) of polymeric elastomers as a function of temperature \(T\) and expansion \(L\) is approximately given by \(f(T, L)=a T\left(L-L_{0}\right)\) where \(a\) and \(L_{0}\) are constants. (a) Use Maxwell's relations to determine the entropy and enthalpy, \(S(L)\) and \(H(L)\), at constant \(T\) and \(p\). (b) If you adiabatically stretch a rubber band by a small amount, its temperature increases, but its volume does not change. Derive an expression for its temperature \(T\) as a function of \(L, L_{0}, a\), and its heat capacity \(C=(\partial U / \partial T)\).

Short answer

At constant T and p, S(L) = \( \frac{a}{2} (L - L_{0})^2 + S_{0} \). H(L) = T \left( \frac{a}{2} (L - L_{0})^2 \). For adiabatic stretching, T(L) = \( T_{0} \text{exp} \left( \frac{a}{2C} (L - L_{0})^2 \right) \).

Step-by-step solution

Analyze the given force equation

The given force equation is: \[ f(T, L) = a T (L - L_{0}) \] where, \[ f = \left( \frac{\partial G}{\partial L} \right)_{T, p} \] and G is the Gibbs free energy. Hence, \[ \left( \frac{\partial G}{\partial L} \right)_{T, p} = a T (L - L_{0}) \]

Use Maxwell relations to find entropy S(L)

From Maxwell's relations, we have: \[ \left( \frac{\partial S}{\partial L} \right)_{T,p} = \left( \frac{\partial^2 G}{\partial T \partial L} \right)_{T,p} \] Since \( \left( \frac{\partial G}{\partial L} \right)_{T, p} = a T (L - L_{0}), \) we differentiate with respect to \( T \) to get: \[ \left( \frac{\partial}{\partial T} \left( \frac{\partial G}{\partial L} \right) \right)_{T, p} = \left( \frac{\partial}{\partial T} (a T (L - L_{0})) \right) = a (L - L_{0}) \] Therefore, \[ \left( \frac{\partial S}{\partial L} \right)_{T, p} = a (L - L_{0}) \] Integrate with respect to L to get S(L): \[ S(L) = a \int (L - L_{0}) dL \] \[ S(L) = \frac{a}{2} (L - L_{0})^2 + S_{0} \]

Use Maxwell's relations to find enthalpy H(L)

From the known relations, \( H = U + pV \) and \( dH = T dS + V dp\). Here, assuming p and V are constants: \[ H - H_{0} = T (S - S_{0}) \] Substitute the derived S(L): \[ H - H_{0} = T \left( \frac{a}{2} (L - L_{0})^2 \right) \] \[ H(L) = H_{0} + T \left( \frac{a}{2} (L - L_{0})^2 \right) \]

Express temperature change due to adiabatic stretching

In adiabatic processes, dQ = 0, hence \( dU = T dS \): \[ T dS = C dT \] Using the previously derived S(L), we have \(T d \left( \frac{a}{2} (L - L_{0})^2 \right) = C dT \): \[ \left( a (L - L_{0}) \right) T dL = C dT \] Solve for dT: \[ dT = \frac{a T (L - L_{0}) dL}{C} \] Integrate both sides to find T: \[ \int_{T_{0}}^{T} \frac{dT}{T} = \int_{L0}^{L} \frac{a (L - L_{0}) dL}{C} \] Solving the integrals: \[ \text{ln}(T) - \text{ln}(T_{0}) = \frac{a}{2C} (L - L_{0})^2 \] Thus, \[ T(L) = T_{0} \text{exp} \left( \frac{a}{2C} (L - L_{0})^2 \right) \]

Key concepts

Maxwell relations

Maxwell relations are a set of equations in thermodynamics that relate different thermodynamic properties through partial derivatives of thermodynamic potentials. They are derived from the fundamental thermodynamic equations and provide a powerful tool for simplifying complex thermodynamic problems.

• They connect quantities like temperature, pressure, volume, and entropy.
• In this exercise, we use the relation \(\left( \frac{\partial S}{\partial L} \right)_{T,p} = \left( \frac{\partial^2 G}{\partial T \partial L} \right)_{T,p}\) to find the entropy change as a function of expansion.
• These relationships help us determine how properties like entropy and enthalpy change with other variables.

Proper understanding of Maxwell relations can simplify the determination of thermodynamic properties without the need for direct measurement.

Enthalpy

Enthalpy (H) is a thermodynamic quantity equivalent to the total heat content of a system. It is defined as \(H = U + pV\), where U is the internal energy, p is the pressure, and V is the volume.
In constant pressure and volume conditions, the change in enthalpy can be directly related to the change in entropy and temperature: \(dH = T dS + V dp\).

• For our polymeric elastomer system, this relation allowed us to determine H(L) from our previously found S(L).
• The derived equation \(H(L) = H_0 + T \left( \frac{a}{2} (L - L_0)^2 \right)\) indicates that the enthalpy of a stretched rubber band depends on its temperature and expansion.

Understanding enthalpy is crucial for processes involving heat transfer, such as in chemical reactions and phase changes.

Entropy

Entropy (S) is a measure of the randomness or disorder of a system. In thermodynamics, an increase in entropy usually corresponds to a more disordered state.
From Maxwell relations, we found the change in entropy for our polymeric elastomer: \(S(L) = \frac{a}{2} (L - L_0)^2 + S_0\).

• This formula indicates that as the rubber band stretches, its entropy increases, reflecting a more disordered state.
• Entropy also influences the energy distribution within the system, affecting how temperature and energy change under different conditions.

Properly understanding entropy allows us to predict the behavior of materials and their responses to external changes, such as temperature or volume variations.

Adiabatic processes

Adiabatic processes are those during which no heat is transferred to or from the system (\(dQ = 0\)).
In an adiabatic process for our rubber band, applying tension without heat exchange increases temperature. Here, the first law of thermodynamics indicates that \(dU = T dS\).
From the given exercise, we derived: \(T(L) = T_0 \text{exp} \left( \frac{a}{2C} (L - L_0)^2 \right)\), showing temperature depends on expansion.

• In adiabatic stretching, the internal energy change results from work done on the system.
• This results in temperature changes that can be predicted using the derived relations.

Adiabatic processes are important in various applications, including engines and atmospheric processes.

Heat capacity

Heat capacity (C) is the amount of heat required to change the temperature of a system by one degree. It is defined as \(C = \left( \frac{\partial U}{\partial T} \right)\).
In the context of our rubber band, it affects how much the temperature changes during adiabatic stretching.

• High heat capacity means the temperature changes slowly with added heat.
• For our rubber band, knowing C helps in calculating the temperature change when it is stretched.

Understanding heat capacity is vital for designing thermal systems and for understanding the thermal properties of materials.