Question

For a gas whose \(p-v-T\) behavior is described by \(Z=1+\) \(B p / R T\), where \(B\) is a function of temperature, derive expressions for the specific enthalpy, internal energy, and entropy changes, \(\left[h\left(p_{2}, T\right)-h\left(p_{1}, T\right)\right],\left[u\left(p_{2}, T\right)-u\left(p_{1}, T\right)\right]\), and \(\left[s\left(p_{2}, T\right)-s\left(p_{1}, T\right)\right]\)

Short answer

Changes in enthalpy, internal energy, and entropy are as follows: \[ h(p_2, T) - h(p_1, T) = RT \ln(\frac{p_2}{p_1} + B(p_2 - p_1) \]\[ u(p_2, T) - u(p_1, T) = RT(\frac{\ln p_2}{p_1}) - 1) \]\[ s(p_2, T) - s(p_1, T) = R\ln(\frac{p_2}{p_1}) \]

Step-by-step solution

- Understand the given equation of state

The given equation of state is expressed as: \[Z = 1 + \frac{Bp}{RT}\]Here, Z is the compressibility factor, B is the temperature-dependent second Virial coefficient, p is the pressure, R is the gas constant, and T is the temperature.

- Define the specific enthalpy, internal energy, and entropy changes

The general forms for the changes in specific enthalpy \(h\), specific internal energy \(u\), and specific entropy \(s\) are needed. These thermodynamic properties for real gases can be found using thermodynamic relations and the compressibility factor Z.

- Derive the expression for specific enthalpy change

Specific enthalpy for a gas can be written as:\[ h = h_{ig} + \, \int_{0}^{p} V dp \]Using the equation of state, we can write the volume V as:\[ V = Z\frac{RT}{p} = \left(1 + \frac{Bp}{RT}\right)\frac{RT}{p} = \frac{RT}{p} + B \]Substituting V into the enthalpy integral, we get:\[ \, \int_{0}^{p} V dp = \int_{0}^{p}\left(\frac{RT}{p} + B\right)dp \]Evaluating the integral gives:\[ \int_{0}^{p}\frac{RT}{p} dp + \int_{0}^{p} B dp = RT \ln\left(\frac{p_{2}}{p_{1}}\right) + B(p_{2} - p_{1}) \]Thus, the change in specific enthalpy is:\[ h(p_{2}, T) - h(p_{1}, T) = RT \ln\left(\frac{p_{2}}{p_{1}}\right) + B(p_{2} - p_{1}) \]

- Derive the expression for specific internal energy change

The specific internal energy is related to specific enthalpy by:\[ u = h - pv \]Using Z, we have:\[ u = h - ZRT \]The change in internal energy \( u(p_2,T) - u(p_1,T) \) becomes:\[ h(p_2,T) - h(p_1,T) - (p_2v_2 - p_1v_1) \]Substitute values,\[ h(p_2,T) - h(p_1,T) - \left(p_2\left(\frac{RT}{p_2} + B\right) - p_1\left(\frac{RT}{p_1} + B\right)\right) \]After simplifying,\[ RT(\ln\left(\frac{p_2}{p_1}\right) - 1) + B(p_2 - p_1) = 0 \]Thus, we get：\[ u(p_2,T) - u(p_1,T) = RT\left(\ln \frac{p_2}{p_1} - 1\right) \]

- Derive the expression for specific entropy change

Using the Maxwell relations, entropy change is given by:\[ ds = \frac{dH}{T} - \frac{dp}{T}\]In terms of Z, we have:\[ \frac{\partial s}{\partial p} \Bigg|_T = \frac{\partial Z}{\partial T} \quad and \quad \frac{\partial H}{\partial p} \Bigg|_T = V - T \frac{\partial V}{\partial T} |p \]The expression becomes:\[ s(p_2, T) - s(p_1, T) = R \ln \Big(\frac{p_2}{p_1} \Big) \]Therefore, the entropy change is:\[ s(p_2, T) - s(p_1, T) = R \ln \left(\frac{p_2}{p_1}\right) \]

Key concepts

Specific Enthalpy Change

Enthalpy, often denoted as h, is a measure of the total energy in a thermodynamic system. It includes internal energy plus the energy required to make space for the gas by displacing its surroundings. Given the equation of state: \[Z = 1 + \frac{Bp}{RT}\] We know specific enthalpy can be expressed as: \[h = h_{ig} + \frac{V}{dp}\] Using the volume V from the equation of state, we can derive the change in specific enthalpy as: \[h(p_2, T) - h(p_1, T) = RT \boldsymbol{\text{ln}}\bigg(\frac{p_2}{p_1}\bigg) + B(p_2 - p_1)\] This equation tells us how enthalpy changes with pressure for real gases. Let's break it down further:

• The term \[RT \boldsymbol{\text{ln}}\bigg(\frac{p_2}{p_1}\bigg)\] represents the contribution due to the ideal gas behavior.
• The term \[B(p_2 - p_1)\] adds a correction for non-ideal behavior, where B is temperature dependent.
This correction accounts for interactions between gas molecules, something ideal gas laws don't cover.
Understanding this will help you apply the concept to various thermodynamic processes involving real gases.

Specific Internal Energy Change

Internal Energy, symbolized by u, is the total energy contained within a system, excluding any energy that is due to the system's position or movement. Using the specific enthalpy, the specific internal energy can be written as: \[u = h - pv\] Substituting the values for h and v, we get: \[u(p_2, T) - u(p_1, T) = h(p_2, T) - h(p_1, T) - (p_2v_2 - p_1v_1)\]After some simplification, this becomes: \[u(p_2, T) - u(p_1, T) = RT \bigg(\boldsymbol{\text{ln}}\bigg(\frac{p_2}{p_1}\bigg) - 1\bigg)\]Notice that once again, it has an ideal part and a part accounting for real gas behavior. This simplified equation helps us understand internal energy changes due to pressure variations.

• The internal energy is related to how gas molecules interact inside the system.
• The term \[RT(\boldsymbol{\text{ln}}\bigg(\frac{p_2}{p_1}\bigg) - 1)\] captures the decrease in internal energy resulting from pressure drops.

Knowing this will aid in understanding the energy dynamics inside real gas systems.

Specific Entropy Change

Entropy, represented by s, measures the degree of disorder or randomness in a system. To find the change in specific entropy, we need Maxwell relations. Using the given state equation and solving for entropy: \[s(p_2, T) - s(p_1, T) = R \boldsymbol{\text{ln}}\bigg(\frac{p_2}{p_1}\bigg)\]This equation is elegant in its simplicity.

• It shows that entropy change is influenced by the ideal gas constant R and the ratio of the pressures.
• No complex correction term is needed for non-ideal behavior in this case, making it easier to calculate compared to enthalpy and internal energy changes.

Entropy changes occur due to heat transfer and are crucial for understanding thermodynamic cycle efficiency.Realizing that entropy depends on pressure variations will help you grasp how real gases behave under different thermodynamic processes, aiding in problem-solving and practical applications.