Question

The \(p-v-T\) relation for a certain gas is represented closely by \(v=R T / p+B-A / R T\), where \(R\) is the gas constant and \(A\) and \(B\) are constants. Determine expressions for the changes in specific enthalpy, internal energy, and entropy, \(\left[h\left(p_{2}, T\right)-\right.\) \(\left.h\left(p_{1}, T\right)\right],\left[u\left(p_{2}, T\right)-u\left(p_{1}, T\right)\right]\), and \(\left[s\left(p_{2}, T\right)-s\left(p_{1}, T\right)\right]\) respectively.

Short answer

\[ h(p_2, T) - h(p_1, T) = (p_2 - p_1) B - \frac{A} {R T}(p_2 - p_1) \]\[ u(p_2, T) - u(p_1, T) = 0 \]\[ s(p_2, T) - s(p_1, T) = -R \ln \left(\frac{p_2}{p_1} \right) \]

Step-by-step solution

Understanding the given relation

The equation given for the gas is: \[ v = \frac{R T}{p} + B - \frac{A}{R T} \]where: - \(v\) is the specific volume,- \(R\) is the gas constant,- \(T\) is the temperature,- \(p\) is the pressure,- \(A\) and \(B\) are constants. Understand that we need to find expressions for changes in specific enthalpy (\(h\)), internal energy (\(u\)), and entropy (\(s\)).

Writing the enthalpy change equation

Specific enthalpy \(h\) is given by: \[ h = u + pv \]Since specific volume \( v \) is known, substitute it in: \[ h = u + p \left( \frac{R T}{p} + B - \frac{A}{R T} \right) \]This simplifies to: \[ h = u + R T + pB - \frac{A p}{R T} \]Since we only need the change in enthalpy, examine \[ \Delta h = h(p_2, T) - h(p_1, T) \].

Simplifying enthalpy change

Using the equation from Step 2, consider the enthalpy at two different pressures but at the same temperature: \[ \Delta h = h_2 - h_1 = \left( u_2 + R T + p_2 B - \frac{A p_2}{R T} \right) - \left( u_1 + R T + p_1 B - \frac{A p_1}{R T} \right) \]This simplifies to: \[ \Delta h = (u_2 - u_1) + (p_2 - p_1)B - \frac{A}{R T} (p_2 - p_1) \]Since we need the enthalpy change only, assume \(u_2 - u_1\) is minimal in cases where only pressure affects enthalpy.

Writing the internal energy change equation

Specific internal energy \(u\) is given by: \[ du = C_v dT + \left( T \frac{\partial p}{\partial T} - p \right) d v \].For change between two pressures at identical temperatures, the integral form results in: \[ \Delta u = u(p_2, T) - u(p_1, T) = \int_{v_1}^{v_2} \left( T \frac{\partial p}{\partial T} - p \right) d v \].

Simplifying internal energy change

The expression for change in volume is: \[ v = \frac{R T}{p} + B - \frac{A}{R T} \]Integrating over pressure changes assuming isothermal conditions: \[ \Delta u = \int_{v1}^{v2} \left( \frac{R T}{v} - p \right) d v \].Usually for minimal changes in relation to volume under constant temperature assumption the result is: \[ u(p_2, T) - u(p_1, T) = 0 \].

Writing the entropy change equation

Specific entropy \(s\) change is: \[ ds = C_p \frac{dT}{T} - R \frac{dp}{p} \].For gas change between temperatures at different pressures: \[ \Delta s = s_2 - s_1 \].

Simplifying entropy change

Under constant temperature consideration: \[ \Delta s = -R \ln \left( \frac{p_2}{p_1} \right) \].Thus, the specific entropy change between two pressures becomes: \[ s(p_2, T) - s(p_1, T) = -R \ln \left( \frac{p_2}{p_1} \right) \].

Key concepts

p-v-T relation

The p-v-T relation is a fundamental concept in gas thermodynamics. It shows the relationship between pressure (p), volume (v), and temperature (T) for a gas. The given relation for a certain gas is expressed as: \[ v = \frac{R T}{p} + B - \frac{A}{R T} \] where:

• v is the specific volume
• R is the gas constant
• T is the temperature
• p is the pressure
• A and B are constants

Understanding this relation is crucial because it aids in deriving how these variables interact and change under different conditions. It sets the stage for calculating other properties like enthalpy, internal energy, and entropy.

specific enthalpy change

Specific enthalpy (\( h \)) is a measure of energy in a thermodynamic system. It combines internal energy (\( u \)) and the product of pressure and volume (\( p v \)). The expression is:\[ h = u + p \left( \frac{R T}{p} + B - \frac{A}{R T} \right) \]. This simplifies to:\[ h = u + R T + pB - \frac{A p}{R T} \]. When looking at the change in enthalpy when moving from one state (\( p_1, T \)) to another (\( p_2, T \)), we get:\[ \Delta h = h(p_2, T) - h(p_1, T) = (u_2 - u_1) + (p_2 - p_1)B - \frac{A}{R T} (p_2 - p_1) \]. This equation is essential for understanding how enthalpy varies, helping in energy balance calculations.

internal energy change

Specific internal energy (\( u \)) is the total energy contained within a system. The differential form is expressed as:\[ du = C_v dT + \left( T \frac{\partial p}{\partial T} - p \right) dv \]. Integrating this, while considering isothermal processes, gives:\[ \Delta u = \int_{v_1}^{v_2} \left( \frac{R T}{v} - p \right) dv \]. However, for practical purposes, under constant temperature cases, the internal energy change (\( \Delta u \)) is often negligible:\[ u(p_2, T) - u(p_1, T) = 0 \]. This concept is critical for simplifying equations in thermodynamic processes.

entropy change

Specific entropy (\( s \)) quantifies the level of disorder or randomness in a thermodynamic system. The change in entropy during a process is given by:\[ ds = C_p \frac{dT}{T} - R \frac{dp}{p} \]. Specially for isothermal processes, this simplifies to:\[ \Delta s = s_2 - s_1 = -R \ln \left( \frac{p_2}{p_1} \right) \]. Entropy change is a pivotal concept in determining the spontaneity and feasibility of processes, playing a key role in the second law of thermodynamics.

thermodynamic equations

Thermodynamic equations are tools for predicting and understanding the behavior of substances in different conditions. Some key equations derived from the gas thermodynamics principles discussed are:

• p-v-T relation: \[ v = \frac{R T}{p} + B - \frac{A}{R T} \]
• Enthalpy change: \[ \Delta h = (u_2 - u_1) + (p_2 - p_1)B - \frac{A}{R T} (p_2 - p_1) \]
• Internal energy change: \[ \Delta u = 0 \] under isothermal conditions
• Entropy change: \[ \Delta s = -R \ln \left( \frac{p_2}{p_1} \right) \]

These equations provide a framework for analyzing and solving thermodynamic problems, enabling deeper insights into energy and material transformations.