Question

At certain states, the \(p-v-T\) data of a gas can be expressed as \(Z=1-A p / T^{4}\), where \(Z\) is the compressibility factor and \(A\) is a constant. (a) Obtain an expression for \((\partial p / \partial T)_{v}\) in terms of \(p, T, A\), and the gas constant \(R\). (b) Obtain an expression for the change in specific entropy, \(\left[s\left(p_{2}, T\right)-s\left(p_{1}, T\right)\right]\) (c) Obtain an expression for the change in specific enthalpy, \(\left[h\left(p_{2}, T\right)-h\left(p_{1}, T\right)\right]\)

Short answer

\( \left( \frac{\partial p}{\partial T} \right)_{v} = R \left( \frac{1}{V + \frac{AV}{T^{3}}} - \frac{AV}{T^{4}} \right) \);\left[ s(p_{2},T) - s(p_{1},T) = -R \ln \left( \frac{p_{2}}{p_{1}} \right) + \frac{AR}{T^{4}}(p_{2}-p_{1}) \]; h(p_{2},T) - h(p_{1},T) = RT\left( -\ln\left( \frac{p_{2}}{p_{1}} \right) + \frac{A}{T^{4}} \right)(p_{2} - p_{1})\.

Step-by-step solution

Understanding the given equation

The compressibility factor is given by the equation: \[Z=1-\frac{A p}{T^{4}}\]where \(Z\) is the compressibility factor, \(A\) is a constant, \(p\) is pressure, and \(T\) is temperature.

Obtain expression for \((\partial p / \partial T)_{v}\)

At constant volume \(v\), the compressibility factor is related to the pressure and temperature. We need to use the equation:\[Z = \frac{pV}{RT}\]where \(V\) is the volume and \(R\) is the gas constant. Now, substitute the given expression for \(Z\):\[1-\frac{A p}{T^{4}} = \frac{pV}{RT}\]Let's rearrange to isolate \(p\):\[pV = RT - \frac{ARp}{T^{3}}V\]\[pV \left(1 + \frac{AV}{T^{3}}\right) = RT\]Finally, solving for \(p\) we get:\[p = \frac{RT}{V + \frac{AV}{T^{3}}}\] Now, differentiate \(p\) with respect to \(T\):\[ \left( \frac{\partial p}{\partial T} \right)_{v} = R \left( \frac{1}{V + \frac{AV}{T^{3}}} - \frac{AV}{T^{4}} \right) \]

Obtain expression for entropy change \(\left[s\left(p_{2}, T\right)-s\left(p_{1}, T\right)\right]\)

Using the Maxwell relations, we know that:\[ \left( \frac{\partial s}{\partial p} \right)_{T} = -\left( \frac{\partial v}{\partial T} \right)_{p}\]Substituting the volume from the given equation, we get:\[ \left( \frac{\partial v}{\partial T} \right)_{p} = \frac{R}{p} - \frac{AR}{T^{4}p}\]Integrate with respect to \(p\) from \(p_{1}\) to \(p_{2}\):\[ s(p_{2},T) - s(p_{1},T) = - \int_{p_{1}}^{p_{2}} \left( \frac{R}{p} - \frac{AR}{T^{4}p} \right) dp = -R \ln\left( \frac{p_{2}}{p_{1}} \right) + \frac{AR}{T^{4}}(p_{2}-p_{1}) \]

Obtain expression for enthalpy change \(\left[h\left(p_{2}, T\right)-h\left(p_{1}, T\right)\right]\)

Using the relationship for enthalpy change:\[ dh = T ds + v dp \]From the equation:\[ v = \frac{RT}{p}(1 - \frac{A}{T^{4}} p) \]Substitute this in the enthalpy change equation and integrate between \(p_{1}\) and \(p_{2}\):\[ h(p_{2},T) - h(p_{1},T) = T \left[-R \ln \left( \frac{p_{2}}{p_{1}} \right) + \frac{AR}{T^{4}} (p_{2} - p_{1}) \right] + \int_{p_{1}}^{p_{2}} \frac{RT}{p}(1 - \frac{A}{T^{4}}p)dp \]Solving this we get:\[ h(p_{2},T) - h(p_{1},T) = RT\left( -\ln\left( \frac{p_{2}}{p_{1}} \right) + \frac{A}{T^{4}} \right)(p_{2} - p_{1}) \]

Key concepts

Partial Derivative of Pressure with Respect to Temperature

In thermodynamics, understanding the relation between pressure (p) and temperature (T) at constant volume (v) is crucial. This is captured by the partial derivative \(\frac{\text{∂} p}{\text{∂} T} \) at constant volume. For a given gas, we know the compressibility factor (Z) is expressed as \[Z = 1 - \frac{A p}{T^4}\text{,}\] where A is a constant. We also use the ideal gas law \[Z = \frac{pV}{RT}\] to relate pressure, volume (V), and the gas constant (R). By substituting the expression for Z and rearranging, we obtain a formula for pressure in terms of T, V, R, and A. From here, we differentiate with respect to T to find the partial derivative of p with respect to T at constant volume. The result helps us understand how pressure changes with temperature under constant volume conditions.

Specific Entropy Change

Entropy (s) is a measure of disorder or randomness in a system. The change in specific entropy when moving between two states at different pressures but constant temperature can be calculated using Maxwell relations. Given \[ \frac{\text{∂} s}{\text{∂} p} = -\frac{\text{∂} v}{\text{∂} T} \] and substituting volume (v) from the given compressibility factor equation, we integrate this expression to find the entropy difference. The result includes two key components: a term involving the natural logarithm of the pressure ratio, and another involving the constant A and temperature. These contributions help us understand how pressure changes influence entropy.

Specific Enthalpy Change

Enthalpy (h) reflects the total heat content of a system. When calculating the specific enthalpy change between two states, we use the relationship: \[ \text{d}h = T \text{d}s + v \text{d}p \] By substituting the expression for volume in terms of temperature, pressure, and the constant A, we integrate to obtain the total change in enthalpy. This provides insight into how thermal energy and work contributions drive changes in system states. The resulting equation comprises terms for both entropic and volumetric contributions, illustrating the complex interplay between energy, pressure, and temperature.

Maxwell Relations

Maxwell relations are a set of equations derived from the fundamental thermodynamic potentials. These relations provide connections between different partial derivatives of thermodynamic quantities. For instance, one of the Maxwell relations tells us: \[ \frac{\text{∂} s}{\text{∂} p} = -\frac{\text{∂} v}{\text{∂} T} \] This relation links entropy (s) to volume (v) and pressure (p) to temperature (T), enabling us to derive expressions for changes in entropy or enthalpy. These relations help simplify calculations in thermodynamics by providing shortcuts to otherwise complex derivations.

Ideal Gas Law

The ideal gas law is a fundamental principle 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 many small particles moving randomly. It provides a simple relationship between pressure, volume, and temperature for ideal gases. Although real gases deviates from this behavior, especially at high pressure and low temperature, the ideal gas law is an essential approximation for understanding gas properties and behaviors under standard conditions.