Question

At certain states, the \(p-v-T\) data for a particular gas can be represented as \(Z=1-A p / T^{4}\), where \(Z\) is the compressibility factor and \(A\) is a constant. Obtain an expression for the specific heat \(c_{P}\) in terms of the gas constant \(R\), specific heat ratio \(k\), and \(Z\). Verify that your expression reduces to Eq. \(3.47 \mathrm{a}\) when \(Z=1\).

Short answer

The specific heat at constant pressure in terms of R, k, and Z is: \[ c_P = R \frac{k}{k-1} Z \]. When Z=1, the expression reduces to the ideal gas specific heat: \[ c_P = \frac{k R}{k - 1} \].

Step-by-step solution

Write the given equation and identify variables

Start with the given equation for the compressibility factor: \[Z = 1 - \frac{A p}{T^4}\]Here, Z is the compressibility factor, A is a constant, p is the pressure, and T is the temperature.

Express specific heat at constant pressure

Use the definition of specific heat at constant pressure, which is related to the compressibility factor (Z). Apply the relation:\[c_P = R \frac{k}{k-1} Z\]where R is the gas constant and k is the specific heat ratio.

Verify the expression for Z=1

When Z = 1 (ideal gas), the expression should reduce to the ideal gas specific heat equation:\[c_P = \frac{k R}{k - 1}\]Insert Z=1 into the general expression to verify this.

Key concepts

Specific Heat at Constant Pressure

Specific heat at constant pressure, denoted as \(c_P\), is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius while maintaining constant pressure. For gases, the specific heat at constant pressure can vary depending on the behavior and state of the gas.
In the given problem, the specific heat at constant pressure is expressed using the gas constant \(R\), the specific heat ratio \(k\), and the compressibility factor \(Z\). This relationship is key for understanding how non-ideal gases differ from ideal gases. The expression given is:
\[c_P = R \frac{k}{k-1} Z \]
This formula accounts for deviations from ideal gas behavior through the compressibility factor \(Z\). It shows how much more (or less) energy is needed due to the gas's deviation from ideal behavior.

Gas Constant

The gas constant, denoted as \(R\), is a fundamental constant in thermodynamics. It appears in many key equations, including the ideal gas law. The value of the gas constant is approximately 8.314 J/(mol·K). It connects the macroscopic properties of gases to their microscopic properties.
The gas constant is essential for calculating the specific heat at constant pressure. In equations related to heat capacities and thermodynamic processes, \(R\) serves as a bridge between the different states and behaviors of gases.
For instance, in the given exercise, the gas constant \(R\) is part of the formula:
\[c_P = R \frac{k}{k-1} Z \]
This highlights its importance in converting between different thermodynamic quantities and ensuring consistency within the calculations.

Specific Heat Ratio

The specific heat ratio, denoted as \(k\) or \(\frac{c_P}{c_V}\), describes the ratio of the specific heat at constant pressure \(c_P\) to the specific heat at constant volume \(c_V\). This ratio is crucial for understanding various thermodynamic processes, such as adiabatic and isentropic processes.
For ideal gases, the specific heat ratio provides insight into how energy is partitioned between potential and kinetic energy. It influences the speed of sound in the gas, the shape of expansion curves, and more.
The provided exercise uses the specific heat ratio in the formula:
\[c_P = R \frac{k}{k-1} Z \]
This relationship shows how \(c_P\) depends not only on the gas’s fundamental properties (as captured by \(R\) and \(k\)) but also on its actual state (as captured by \(Z\)).

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics, expressed as \[PV = nRT\], where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature. This law assumes that gases consist of non-interacting particles and that the volume of these particles is negligible.
In real-life applications, gases often deviate from this ideal behavior, particularly at high pressures and low temperatures. The compressibility factor \(Z = 1 - \frac{A p}{T^4}\) in the exercise measures these deviations. When \(Z = 1\), the gas behaves ideally, and the specific heat at constant pressure simplifies to:
\[c_P = \frac{k R}{k - 1}\]
This shows consistency with the ideal gas law. As \(Z\) varies from 1, we understand better how actual gases behave differently than predicted by the ideal gas law, and how this affects measurements like specific heat.