Problem 7 - A Modern Course in Statistical Physics

Question

Experimentally one finds that for a rubber band $$ \begin{aligned} &\left(\frac{\partial J}{\partial L}\right)_{\mathrm{T}, M}=\frac{a T}{L_{0}}\left[1+2\left(\frac{L_{0}}{L}\right)^{3}\right] \quad \text { and } \\\ &\left(\frac{\partial J}{\partial T}\right)_{L, M}=\frac{a L}{L_{0}}\left[1-\left(\frac{L_{0}}{L}\right)^{3}\right] \end{aligned} $$ where \(J\) is the tension, \(a=1.0 \times 10^{3} \mathrm{dyn} / \mathrm{K}\), and \(L_{0}=0.5 \mathrm{~m}\) is the length of the band when no tension is applied. The mass \(M\) of the rubber band is held fixed. (a) Compute \((\partial L / \partial T)_{J, M}\) and discuss its physical meaning. (b) Find the equation of state and show that \(\mathrm{d} J\) is an exact differential. (c) Assume that the heat capacity at constant length is \(C_{L}=1.0 \mathrm{~J} / \mathrm{K}\). Find the work necessary to stretch the band reversibly and adiabatically to a length of \(1 \mathrm{~m}\). Assume that when no tension is applied, the temperature of the band is \(T=290 \mathrm{~K}\). What is the change in temperature?

Step-by-Step Solution

Answer

For part (a), \(\left(\frac{\partial L}{\partial T}\right)_{J, M} = \left[\frac{a T}{L_0}\left(1+2\left(\frac{L_0}{L}\right)^3\right)\right]^{-1}\). Physically, it represents the thermal expansion coefficient. The equation of state and exact differential need detailed integration. Finally, compute the work and temperature change in the adiabatic process.

Step 1: Determine \((\(\partial L/\partial T\))_{J, M}\)

Given the partial derivatives, use the Maxwell relation from thermodynamics: \(\left(\frac{\partial J}{\partial L}\right)_{T, M} = \left(\frac{\partial T}{\partial J}\right)_{L, M}\). The expression given is \(\left(\frac{\partial J}{\partial L}\right)_{T, M}=\frac{a T}{L_{0}}\left[1+2\left(\frac{L_{0}}{L}\right)^{3}\right]\). By inverting this, \(\left(\frac{\partial L}{\partial T}\right)_{J, M} = \left[\frac{a T}{L_{0}}\left(1+2\left(\frac{L_{0}}{L}\right)^{3}\right)\right]^{-1}\).

Step 2: Discuss the physical meaning

The partial derivative \((\frac{\partial L}{\partial T})_{J, M}\) indicates how the length of the rubber band changes with temperature when the tension and mass are held constant. It is the thermal expansion coefficient under constant tension.

Step 3: Find the equation of state

Combine the given expressions to form the state equation. From \(\left(\frac{\partial J}{\partial L}\right)_{T, M}\) and \(\left(\frac{\partial J}{\partial T}\right)_{L, M}\), solve for \(J\) as a function of \(L\) and \(T\).

Step 4: Show that \(dJ\) is an exact differential

Verify exactness by checking the equality \(\left(\frac{\partial^2 F}{\partial T \partial L}\right) = \left(\frac{\partial^2 F}{\partial L \partial T}\right)\). Use the given partial derivatives and confirm their consistency.

Step 5: Compute work and temperature change

In an adiabatic reversible process, use the expression for work, \(W = \int J dL\), given \(C_L = 1.0 \mathrm{J} / \mathrm{K} \). Integrate the tension \(J\) from initial to final length. The change in temperature \(\Delta T\) can be found using the first law of thermodynamics, \(dU = TdS - PdV\).

Step 6: Apply initial conditions and solve

With initial length \(L_0 = 0.5\) m, initial temperature \(T = 290 \) K, and the final length \(L = 1.0\) m, evaluate the integrals and compute the temperature change during the adiabatic process.

Key Concepts

Maxwell Relations

Maxwell relations are a set of equations derived from the thermodynamic potentials. They help us connect different partial derivatives that describe physical properties like temperature, pressure, volume, and entropy.
These relations originate from the second law of thermodynamics, specifically the exact differentials of the internal energy, Helmholtz free energy, Gibbs free energy, and enthalpy.

For this problem, we use the Maxwell relation \((\( \left( \frac{\partial L}{\partial T} \right)_{J, M} = \left( \frac{a T}{L_{0}}\left(1+2\left( \frac{L_{0}}{L} \right)^{3} \right)\)^{-1} \)). This relation tells us how the length changes with temperature when the tension and mass are constant.

Using these relations simplifies the computation and increases our understanding of thermodynamic systems.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature.
In the case of rubber bands, they expand when heated and contract when cooled.
In this exercise, the term \( \left( \frac{\partial L}{\partial T} \right)_{J, M} \) gives the thermal expansion coefficient under constant tension.

Essentially, it suggests how much the length of the rubber band will change with a given change in temperature while tension remains constant.
This coefficient is crucial in designing systems where thermal expansion needs consideration, such as in materials engineering and physics.

Equation of State

The equation of state describes the relationship between various state variables like temperature, pressure, and volume for a given substance.
It helps in predicting the state of a system under different conditions.
For the rubber band, we combine the expressions
\( \left( \frac{\partial J}{\partial L} \right)_{T, M} = \frac{a T}{L_{0}} \left[ 1+2 \left( \frac{L_{0}}{L} \right)^{3}\right] \) and
\( \left( \frac{\partial J}{\partial T} \right)_{L, M} = \frac{a L}{L_{0}} \left[ 1 \ - \left( \frac{L_{0}}{L} \right)^{3} \right] \).

This joined expression captures the state of the band concerning its length and temperature.
It assists in explaining how the rubber band's tension varies with temperature and length.

Exact Differential

In thermodynamics, an exact differential is one where the mixed partial derivatives of a function are equal.
This condition ensures that the function is well-defined and path-independent.

For the rubber band, we verify exactness by confirming the equality
\( \left( \frac{ \partial^2 F}{\partial T \partial L} \right) = \left( \frac{ \partial^2 F}{\partial L \partial T} \right) \).
Here, we use the given partial derivatives and check their consistency to determine if \( dJ \) is an exact differential.

If they match, it means that the expression we have for the tension is consistent and valid throughout the state space of our thermodynamic system.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings.
It is a key concept in thermodynamics and heavily referenced in this problem.

During an adiabatic process, any energy change in the system results from work done on or by the system.
For the rubber band stretched reversibly and adiabatically, the work done is \( W = \int J \, dL \).
By integrating the tension from initial to final length, we can compute the work required.

This understanding allows us to determine the change in temperature of the rubber band, leveraging the first law of thermodynamics.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation.
It declares that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.

Mathematically, it is expressed as \( dU = TdS - PdV \).
In our rubber band problem, considering an adiabatic process, \( dU = - PdV \), translates into understanding how internal energy shifts as the band stretches.

With an initial temperature of 290 K and final length of 1.0 m, applying this law helps us compute the temperature change during the stretching process.

The heat capacity at constant length, \( C_L = 1.0 \) J/K, further aids in solving this exercise, ensuring all calculations align perfectly under the thermodynamic principles.