Question

The Helmholtz function of a substance has the form $$a=-R T \ln \frac{v}{v_{0}}-c T_{0}\left(1-\frac{T}{T_{0}}+\frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right)$$ where \(T_{0}\) and \(v_{0}\) are the temperature and specific volume at a reference state. Show how to obtain \(P, h, s, c_{v}\) and \(c_{p}\) from this expression.

Short answer

#Answer# The expressions for pressure (P), specific enthalpy (h), specific entropy (s), specific heat at constant volume (c_v), and specific heat at constant pressure (c_p) can be found as follows: 1. Pressure (P): $$ P = \frac{RT}{v}. $$ 2. Specific enthalpy (h): $$ h = -c T_{0} \left(1-\frac{T}{T_{0}}+\frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right)+Ts. $$ 3. Specific entropy (s): $$ s = R \ln \frac{v}{v_{0}} + c T_{0} \left(\frac{T}{T_{0}} \ln \frac{T}{T_{0}} - 1-\ln \frac{T}{T_{0}}\right). $$ 4. Specific heat at constant volume (c_v): $$ c_v = -c + R \left(1-\ln \frac{T}{T_{0}}\right). $$ 5. Specific heat at constant pressure (c_p): $$ c_p = R(2 - \ln\frac{T}{T_{0}}) - c. $$

Step-by-step solution

Find the pressure (P)

Differentiate the given expression of a with respect to specific volume (v) at constant temperature (T), and negate the result to find the expression for P:$$ P = -\left(\frac{\partial a}{\partial v}\right)_T = -R T \left(-\frac{1}{v}\right) = \frac{RT}{v}. $$

Find the specific enthalpy (h)

First, differentiate the given expression of a with respect to temperature (T) at constant specific volume (v) and negate the result to find the expression for specific entropy (s):$$ s = -\left(\frac{\partial a}{\partial T}\right)_v = R \ln \frac{v}{v_{0}} + c T_{0} \left(\frac{T}{T_{0}} \ln \frac{T}{T_{0}} - 1-\ln \frac{T}{T_{0}}\right). $$Then, find the specific internal energy (u) using the relationship between Helmholtz function (a) and specific entropy (s),$$ u = a + Ts = -R T \ln \frac{v}{v_{0}} - c T_{0} \left(1 - \frac{T}{T_{0}} + \frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right) + Ts. $$Finally, find the specific enthalpy (h) using the definition of enthalpy,$$ h = u + Pv = -R T \ln \frac{v}{v_{0}} - c T_{0} \left(1 - \frac{T}{T_{0}} + \frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right) + Ts + \frac{RT}{v}v = -c T_{0} \left(1-\frac{T}{T_{0}}+\frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right)+Ts. $$

Find the specific heat at constant volume (c_v)

Differentiate the expression of specific internal energy (u) with respect to temperature (T) at constant specific volume (v) to find the expression of c_v:$$ c_v = \left(\frac{\partial u}{\partial T}\right)_v = -c + R \left(1-\ln \frac{T}{T_{0}}\right). $$

Find the specific heat at constant pressure (c_p)

Use Mayer's relation to find the expression for c_p, which relates c_v and c_p as$$ c_p = c_v + R = -c + R \left(1-\ln \frac{T}{T_{0}}\right) + R = R(2 - \ln\frac{T}{T_{0}}) - c. $$Now, we have derived expressions for Pressure (P), Specific enthalpy (h), Specific entropy (s), Specific heat at constant volume (c_v), and Specific heat at constant pressure (c_p).

Key concepts

Thermodynamic Properties

Understanding thermodynamic properties is crucial to the study of how substances interact with energy and with each other. These properties, which are often referred to as state functions, include internal energy (U), enthalpy (H), entropy (S), and specific volume (v), among others. They describe the state of a thermodynamic system independently of how the system reached that state.

For example, the internal energy represents the total energy within a system, encompassing kinetic energy, potential energy, and chemical energy. Entropy, on the other hand, is a measure of the disorder or randomness of a system and is essential when discussing the second law of thermodynamics. The Helmholtz function, which was the focal point of the exercise, directly relates to these properties and is particularly useful for determining equilibrium in closed systems at constant temperature and volume.

Specific Heat Capacities

Specific heat capacities, symbolized as c_v and c_p for constant volume and constant pressure respectively, are measures of the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius (or one Kelvin). The specific heat at constant volume, c_v, is the energy required to raise the temperature of a unit mass of a substance by one degree while maintaining constant volume. In contrast, the specific heat at constant pressure, c_p, accounts for the work done when a substance expands.

The difference between these two capacities is fundamental to understanding thermodynamic processes and can be explained by how the substance's volume can change when heated under constant pressure conditions. The exercise showcased the importance of specific heat capacities in calculating changes in thermodynamic quantities, such as enthalpy and specific entropy.

Enthalpy and Internal Energy

Enthalpy (H) and internal energy (U) are both energy measures within a thermodynamic system but differ in their application. Enthalpy is the sum of internal energy and the product of pressure and volume (\(H = U + PV\)), making it useful in processes happening at constant pressure. It represents the total heat content of a system and is often involved in describing heat transfer during chemical reactions and phase changes.

Internal energy, as mentioned earlier, accounts for all energy contained in a substance, including translational, rotational, and vibrational kinetic energies of particles, as well as potential energy from intermolecular forces. The solution to the exercise demonstrated how to derive expressions for both enthalpy, which includes the PV term, and internal energy from the Helmholtz function, highlighting their interrelation and relevance to thermodynamics.

Mayer's Relation

Mayer's relation is an essential concept in thermodynamics, providing a link between the specific heats at constant pressure (c_p) and constant volume (c_v) through the gas constant (R). It is succinctly described by the formula \( c_p - c_v = R \), which is applicable to ideal gases where the internal energy depends solely on temperature.

This relationship tells us that the specific heat at constant pressure, c_p, is always larger than c_v because work is done by the system when it expands against an external pressure. The given exercise uses Mayer's relation to derive an expression for c_p from the obtained value of c_v. This solidifies the understanding that Mayer’s relation is central for analyzing thermodynamic processes and calculating the work and heat involved.