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Chapter 3: Problem 98

Five \(\mathrm{kg}\) of butane \(\left(\mathrm{C}_{4} \mathrm{H}_{10}\right)\) in a piston-cylinder assembly undergoes a process from \(p_{1}=5 \mathrm{MPa}, T_{1}=500 \mathrm{~K}\) to \(p_{2}=3 \mathrm{MPa}\), during which the relationship between pressure and specific volume is \(p v=\) constant. Determine the work, in kJ.

Short Answer

Expert verified
6.17 kJ

Step by step solution

01

Understand the Given Data

Identify and list the given information: initial pressure \(p1 = 5 \space MPa\), initial temperature \(T1 = 500 \space K\), final pressure \(p2 = 3 \space MPa\), mass of substance \(m = 5 \space kg\), and the relationship \(pv = constant\).
02

Express the Polytropic Process

For a polytropic process where \(pv = constant\), we know that \(p \times v^n = constant\). Here, \(n = 1\), so \(p1 \times v1 = p2 \times v2 \). This will be used to find the specific volume.
03

Calculate Initial Specific Volume

Use the ideal gas law \(pv = RT\) to find the initial specific volume \(v1\): \(v1 = \frac{RT1}{p1}\). Use \(R = 0.143 \space kJ/(kg\cdot K)\) for butane. \[ v1 = \frac{0.143 \times 500}{5} = 0.0143 \space m^3/kg \]
04

Calculate Final Specific Volume

Since \(p1 \times v1 = p2 \times v2 \), we can solve for \(v2\): \[ v2 = \frac{p1 \times v1}{p2} = \frac{5 \times 0.0143}{3} = 0.02383 \space m^3/kg \]
05

Determine the Work Done

For a polytropic process (with \(n = 1\)), the work done is given by: \[ W = \frac{p1 \times v1 - p2 \times v2}{1 - n} \] \ Since \(n = 1,\), use the logarithmic form: \[ W = p1 \times V1 \times ln\frac{V2}{V1} \] \ Remember that total volumes \(V1\) and \(V2\) can be found by multiplying the specific volumes by mass \(m\): \[ W = p1 \times (m \times v1) \times ln\frac{m \times v2}{m \times v1} = p1 \times m \times v1 \times ln\frac{v2}{v1} \]
06

Calculate the Work Done

Substitute the known values: \[ W = 5 \times 5 \times 0.0143 \times ln\frac{0.02383}{0.0143} = 5 \times 5 \times 0.0143 \times ln(1.667) \] Use the value of \(ln(1.667) 鈮� 0.5108 \): \[ W 鈮� 5 \times 5 \times 0.0143 \times 0.5108 = 0.18267 \times 2.554 鈮� 6.166 \space kJ \]

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Key Concepts

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

Ideal Gas Law
The Ideal Gas Law is a fundamental equation in thermodynamics. It relates the pressure, volume, and temperature of an ideal gas with the number of moles and the gas constant. The formula is: \[ PV = nRT \] In this equation:
  • P: Pressure of the gas
  • V: Volume of the gas
  • n: Number of moles of the gas
  • R: Universal gas constant (8.314 J/(mol.K))
  • T: Temperature of the gas in Kelvin
For butane, the specific gas constant is used, which is specific to its properties and ensures the equation remains accurate for this particular gas. In the exercise, we used the specific gas constant for butane, which is 0.143 kJ/(kg路K), and substituted it into the Ideal Gas Law to find the initial specific volume.
Specific Volume
Specific volume is the volume per unit mass of a substance. For gases, it represents how much space a specific amount of gas occupies. It is expressed in cubic meters per kilogram (m鲁/kg).
Given the Ideal Gas Law, specific volume can be calculated as: \[ v = \frac{RT}{P} \] In the context of the exercise, this formula was used to find the initial specific volume (\( v1 \)) of butane. Using the initial conditions (P1 = 5 MPa, T1 = 500 K), the specific gas constant for butane (R = 0.143 kJ/(kg路K)), we substituted into the formula to get: \[ v1 = \frac{0.143 \times 500}{5} = 0.0143 \ space m^3/kg \]
Work Done in Thermodynamic Processes
Work done in a thermodynamic process is the energy transferred when a gas expands or contracts under pressure. For different processes (isothermal, adiabatic, polytropic), the work done can be calculated using different formulas.
  • In an isothermal process, the work done is given by: \[ W = P \times V \times ln\left( \frac{V2}{V1} \right) \]
In our exercise, the process follows a polytropic relationship with n=1 (which models an isothermal process), where pressure and volume follow the law \( PV^n = \text{constant} \).
To find the total work done, we used the equation: \[ W = P1 \times V1 \times ln\left( \frac{V2}{V1} \right) \]Substituting the values: \[ V1 = m \times v1, \quad V2 = m \times v2 \] we calculated: \[ W 鈮� 6.166 \text{kJ} \]
Butane Properties
Butane (\( C4H10 \)) is a hydrocarbon commonly used as a fuel. It's important to understand its thermodynamic properties to accurately apply formulas and compute work done during processes. Some key properties include:
  • Specific gas constant (\( R = 0.143 \text{kJ/(kg路K)} \))
  • Molecular weight: 58.12 g/mol
  • Critical temperature and pressure: 425.2 K and 3.8 MPa respectively
In this exercise, butane's properties, particularly its specific gas constant, were crucial for accurate calculations.
Pressure-Volume Relationship
The pressure-volume relationship in gases is fundamental to understanding thermodynamic processes. In our exercise, the process follows a polytropic relationship (\( PV^n = \text{constant} \)).
For an isothermal process, we see that: \[ P1 \times V1 = P2 \times V2 \] This relationship helped us find the final specific volume (\( v2 \)) by knowing the initial conditions and the final pressure. Specifically, rearranging the equation, we find:
  • \[ V2 = \frac{P1 \times V1}{P2} \]
Substituting our values, we were able to determine the final specific volume and calculate the work done.

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Most popular questions from this chapter

Air contained in a piston-cylinder assembly, initially at 2 bar, \(200 \mathrm{~K}\), and a volume of \(1 \mathrm{~L}\), undergoes a process to a final state where the pressure is 8 bar and the volume is \(2 \mathrm{~L}\). During the process, the pressure-volume relationship is linear. Assuming the ideal gas model for the air, determine the work and heat transfer, each in kJ.

An open container of pure ethanol (ethyl alcohol) liquid is placed on a table in a room. Evaporation occurs until all of the ethanol is gone. Where did the ethanol go? If the ethanol and the room air are taken to be a closed system, can the system be regarded as a pure substance during the process? How many phases are present initially and finally? Explain. Using \(p-v-T\) Data

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) contained in a piston-cylinder arrangement, initially at 6 bar and \(400 \mathrm{~K}\), undergoes an expansion to a final temperature of \(298 \mathrm{~K}\), during which the pressure-volume relationship is \(p V^{1.2}=\) constant. Assuming the ideal gas model for the \(\mathrm{CO}_{2}\), determine the final pressure, in bar, and the work and heat transfer, each in \(\mathrm{kJ} / \mathrm{kg}\).

Two kg of oxygen fills the cylinder of a piston-cylinder assembly. The initial volume and pressure are \(2 \mathrm{~m}^{3}\) and 1 bar, respectively. Heat transfer to the oxygen occurs at constant pressure until the volume is doubled. Determine the heat transfer for the process, in kJ, assuming the specific heat ratio is constant, \(k=1.35\). Kinetic and potential energy effects can be ignored.

A piston-cylinder assembly contains air, initially at 2 bar, \(300 \mathrm{~K}\), and a volume of \(2 \mathrm{~m}^{3}\). The air undergoes a process to a state where the pressure is 1 bar, during which the pressure-volume relationship is \(p V=\) constant. Assuming ideal gas behavior for the air, determine the mass of the air, in \(\mathrm{kg}\), and the work and heat transfer, each in \(\mathrm{kJ}\).
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