Consider the ideal elastic, whose equation or state is given...

Consider the ideal elastic, whose equation or state is given by $(0.20)$. (i) Show that the internal energy is a function of temperature alone (in this respect the ideal elastic substance is like an ideal gas. It is a reflection of the fact that the energy is associated with kinetic not potential energy of the molecules). (ii) If the heat capacity $C_L$ is independent of temperature at $L=L_0$, show that the Helmholtz function is given by $$ F=F_0-T C_L \ln \frac{T}{T_0}+K T\left[\frac{L^2}{2 L_0}+\frac{L_0^2}{L}\right], $$ where $F_0(T)=U_0-T S_0+\left(T-T_0\right) C_L-(3 / 2) L_0 K T$. [Use equation (13.53), suitably modified.] (iii) Show that the difference between the primary heat capacities is $$ C_L-C_f=-\frac{K\left(L^3-L_0^3\right)^2}{L_0 L\left(L^3+2 L_0^3\right)} $$

Consider the ideal elastic, whose equation or state is given by $(0.20)$.
(i) Show that the internal energy is a function of temperature alone (in this respect the ideal elastic substance is like an ideal gas. It is a reflection of the fact that the energy is associated with kinetic not potential energy of the molecules).
(ii) If the heat capacity $C_L$ is independent of temperature at $L=L_0$, show that the Helmholtz function is given by
$$
F=F_0-T C_L \ln \frac{T}{T_0}+K T\left[\frac{L^2}{2 L_0}+\frac{L_0^2}{L}\right],
$$
where $F_0(T)=U_0-T S_0+\left(T-T_0\right) C_L-(3 / 2) L_0 K T$. [Use equation (13.53), suitably modified.]
(iii) Show that the difference between the primary heat capacities is
$$
C_L-C_f=-\frac{K\left(L^3-L_0^3\right)^2}{L_0 L\left(L^3+2 L_0^3\right)}
$$

Key Concepts

Thermodynamic Potentials and Legendre Transformations

Thermodynamic potentials, like the Helmholtz free energy, are central to describing the equilibrium and response properties of thermodynamic systems. The Legendre transformation is a mathematical tool used to switch between different sets of natural variables (e.g., from entropy to temperature) in the analysis of these systems. This transformation enables one to obtain new potentials that reveal different aspects of the system’s behavior, such as stability under constant temperature or constant deformation conditions.

Helmholtz Free Energy

The Helmholtz free energy is a thermodynamic potential that represents the useful work obtainable from a closed system at constant temperature and volume. It is defined through a Legendre transformation, linking internal energy and entropy, and provides insight into the equilibrium properties of the system. In this problem, deriving the explicit form of the Helmholtz free energy helps in understanding the energetics of the system under elastic deformations.

Differences between Primary Heat Capacities

Primary heat capacities refer to the measures of heat needed to raise the temperature of a system under different constraints (such as constant length or constant applied force). The difference between these heat capacities provides information on how the system's response to heating is influenced by its mechanical or elastic properties. This concept illustrates the coupling between mechanical work and thermal effects, a fundamental aspect of thermodynamic systems with elastic behavior.

Internal Energy Dependence on Temperature

In many thermodynamic systems, especially those where the energy is primarily kinetic, the internal energy is determined solely by the temperature. This key concept highlights that the microscopic motions (kinetic energies) of the molecules, rather than any potential interactions, are what contribute to the system's internal energy. This behavior is notably similar to that of an ideal gas, where the internal energy is a direct function of temperature only.

Heat Capacity

Heat capacity is a measure of the amount of heat required to change a system's temperature by a given amount. In this context, the use of a constant heat capacity (independent of temperature) simplifies the analysis of energy changes in the system. This assumption is often employed to obtain explicit expressions for thermodynamic potentials and to explore how energy is partitioned in the system.

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