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Chapter 11: Problem 37

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

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

Step by step solution

01

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 (\(s\)).
02

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 \[ \Delta h = h(p_2, T) - h(p_1, T) \].
03

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.
04

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: \[ \Delta u = u(p_2, T) - u(p_1, T) = \int_{v_1}^{v_2} \left( T \frac{\partial p}{\partial T} - p \right) d v \].
05

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: \[ u(p_2, T) - u(p_1, T) = 0 \].
06

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: \[ \Delta s = s_2 - s_1 \].
07

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: \[ s(p_2, T) - s(p_1, T) = -R \ln \left( \frac{p_2}{p_1} \right) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right.\) ) enters a turbine operating at steady state at 100 bar, \(400 \mathrm{~K}\) and expands isothermally without irreversibilities to 10 bar. There are no significant changes in kinetic or potential energy. Using data from the generalized fugacity chart, determine the power developed, in \(\mathrm{kW}\), for a mass flow rate of \(50 \mathrm{~kg} / \mathrm{min}\).

Derive an equation for the Joule-Thomson coefficient as a function of \(T\) and \(v\) for a gas that obeys the van der Waals equation of state and whose specific heat \(c_{v}\) is given by \(c_{v}=\) \(A+B T+C T^{2}\), where \(A, B, C\) are constants. Evaluate the temperatures at the inversion states in terms of \(R, v\), and the van der Waals constants \(a\) and \(b\).

In the expression \((\partial u / \partial T)_{v}\), what does the subscript \(v\) signify?

During a phase change from liquid to vapor at fixed pressure, the temperature of a binary nonazeotropic solution such as an ammonia-water solution increases rather than remains constant as for a pure substance. This attribute is exploited in both the Kalina power cycle and in the Lorenz refrigeration cycle. Write a report assessing the status of technologies based on these cycles. Discuss the principal advantages of using binary nonazeotropic solutions. What are some of the main design issues related to their use in power and refrigeration systems?

If the function \(p=p(T, v)\) is an equation of state, is \((\partial p / \partial T)_{v}\) a property? What are the independent variables of \((\partial p / \partial T)_{v} ?\)
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