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Chapter 15: Problem 20

Six grams of helium (molecular mass \(=4.0 \mathrm{u}\) ) expand isothermally at \(370 \mathrm{K}\) and \(\mathrm{do} 9600 \mathrm{J}\) of work. Assuming that helium is an ideal gas, determine the ratio of the final volume of the gas to the initial volume.

Short Answer

Expert verified
The final volume is approximately 7.754 times the initial volume.

Step by step solution

01

Determine the Number of Moles

To find the number of moles \( n \) of helium, we use the formula \( n = \frac{m}{M} \), where \( m \) is the mass of helium and \( M \) is its molar mass (4 g/mol). Since the given mass is 6 g, we calculate: \[ n = \frac{6}{4} = 1.5 \text{ moles} \].
02

Apply the First Law of Thermodynamics

For isothermal expansion, the change in internal energy \( \Delta U = 0 \). Therefore, the work done \( W = Q \) (heat added). Given that \( W = 9600 \text{ J} \), this value also represents the heat exchanged.
03

Use the Ideal Gas Law for Isothermal Process

For an isothermal process involving an ideal gas, the work done is given by the formula: \( W = nRT \ln\left( \frac{V_f}{V_i} \right) \). Here, \( n = 1.5 \text{ moles}, \) \( R = 8.314 \text{ J/mol K}, \) \( T = 370 \text{ K}, \) and \( W = 9600 \text{ J} \).
04

Solve for Volume Ratio

Rearrange the work formula to find the volume ratio: \[ \ln\left( \frac{V_f}{V_i} \right) = \frac{W}{nRT} \]. Substitute the numerical values: \[ \ln\left( \frac{V_f}{V_i} \right) = \frac{9600}{1.5 \times 8.314 \times 370} \approx 2.048 \].
05

Exponentiate to Find the Ratio

Raise \( e \) to the power of the result from the previous step to solve for \( \frac{V_f}{V_i} \): \[ \frac{V_f}{V_i} = e^{2.048} \approx 7.754 \].

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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 crucial for understanding the behavior of gases under various conditions. It's expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (8.314 \( \text{J/mol K} \)), and \( T \) is the temperature in Kelvin. This equation ties these variables together by representing the state of an "ideal" gas.
For an isothermal process, like the one in our example where helium gas expands, the temperature \( T \) remains constant. As a result, any change in volume \( V \) must be accompanied by a corresponding change in pressure \( P \), or the amount of work done. The work done by the gas can be calculated using the formula \( W = nRT \ln\left( \frac{V_f}{V_i} \right) \).
Here, knowing the work done and maintaining an isothermal condition, we can use this specific form of the Ideal Gas Law to find the ratio of the final volume \( V_f \) to the initial volume \( V_i \), a critical step in understanding the expansion process.
First Law of Thermodynamics
The First Law of Thermodynamics is a fundamental principle stating that energy cannot be created or destroyed, only transformed. It is often expressed in the form \( \Delta U = Q - W \).
For isothermal processes, like the expansion of helium gas, the internal energy change (\( \Delta U \)) for an ideal gas is zero because the temperature remains constant. This simplifies the First Law to \( Q = W \), meaning all work done by the gas comes from the heat added to it.
This principle explains how and why energy is conserved in the process of isothermal expansion. Understanding this energy transformation is key to solving problems involving work and heat in thermodynamic processes.
Helium Gas
Helium is a noble gas with unique properties that make it behave nearly perfectly as an ideal gas. It's represented by the chemical symbol He and has an atomic mass unit of approximately 4.0. This means that helium is composed of monatomic particles, simplifying calculations involving its thermodynamic behavior.
In our example, helium undergoes an isothermal expansion, and we need to determine the number of moles. With 6 grams of helium, using its molar mass (4 g/mol), we find there are 1.5 moles of helium gas involved.
  • Since helium behaves ideally, it's easy to apply the Ideal Gas Law and First Law of Thermodynamics.
  • Helium's stability and simplicity as an individual atom make it an excellent choice for many practical and theoretical applications in physics and engineering.
Because of this behavior, helium gas often serves as a model for demonstrating the principles of thermodynamics.

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Most popular questions from this chapter

Each of two Carnot engines uses the same cold reservoir at a temperature of \(275 \mathrm{K}\) for its exhaust heat. Each engine receives \(1450 \mathrm{J}\) of input heat. The work from either of these engines is used to drive a pulley arrangement that uses a rope to accelerate a \(125-\mathrm{kg}\) crate from rest along ahorizontal frictionless surface, as shown in the figure. With engine 1 the crate attains a speed of \(2.00 \mathrm{m} / \mathrm{s},\) while with engine 2 it attains a speed of \(3.00 \mathrm{m} / \mathrm{s}\) Concepts: (i) With which engine is the change in the crate's energy greater? (ii) Which engine does more work? Explain your answer. (iii) For which engine is the temperature of the hot reservoir greater? Calculations: Find the temperature of the hot reservoir for each engine.

Carnot engine A has an efficiency of \(0.60,\) and Carnot engine \(\mathrm{B}\) has an efficiency of 0.80. Both engines utilize the same hot reservoir, which has a temperature of \(650 \mathrm{K}\) and delivers \(1200 \mathrm{J}\) of heat to each engine. Find the magnitude of the work produced by each engine and the temperatures of the cold reservoirs that they use.

A Carnot heat pump operates between an outdoor temperature of \(265 \mathrm{K}\) and an indoor temperature of \(298 \mathrm{K}\). Find its coefficient of performance.

The sun is a sphere with a radius of \(6.96 \times 10^{8} \mathrm{m}\) and an average surface temperature of \(5800 \mathrm{K}\). Determine the amount by which the sun's thermal radiation increases the entropy of the entire universe each second. Assume that the sun is a perfect blackbody, and that the average temperature of the rest of the universe is \(2.73 \mathrm{K}\). Do not consider the thermal radiation absorbed by the sun from the rest of the universe.

A gas, while expanding under isobaric conditions, does 480 J of work. The pressure of the gas is \(1.6 \times 10^{5} \mathrm{Pa}\), and its initial volume is \(1.5 \times\) \(10^{-3} \mathrm{m}^{3} .\) What is the final volume of the gas?
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