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Chapter 20: Problem 4

How much energy must be transferred as heat for a reversible isothermal expansion of an ideal gas at \(132^{\circ} \mathrm{C}\) if the entropy of the gas increases by \(46.0 \mathrm{~J} / \mathrm{K}\) ?

Short Answer

Expert verified
The energy transferred as heat is 18636.9 J.

Step by step solution

01

Understand the Question

We need to find the amount of energy transferred as heat during a reversible isothermal expansion of an ideal gas. The process is isothermal, meaning the temperature remains constant.
02

Convert Temperature to Kelvin

The given temperature is in degrees Celsius. To convert it to Kelvin, use the formula: \[ T(K) = T(^{\circ}C) + 273.15 \]For this problem, \( T = 132 + 273.15 = 405.15 \text{ K} \).
03

Use the Relationship between Heat and Entropy

For a reversible process, the change in entropy \( \Delta S \) and heat \( Q \) are related by:\[ Q = T \Delta S \]where \( T \) is the absolute temperature in Kelvin and \( \Delta S \) is the change in entropy. In this situation, \( \Delta S = 46.0 \text{ J/K} \).
04

Calculate the Energy Transferred as Heat

Substitute the values into the equation from Step 3:\[ Q = 405.15 \times 46.0 \]Calculate \( Q \).
05

Solve for Q

Perform the calculation from Step 4:\[ Q = 405.15 \times 46.0 = 18636.9 \text{ J} \]

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Key Concepts

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

Ideal Gas
An ideal gas is a theoretical gas composed of a set of randomly moving, non-interacting point particles. It's a model used to simplify the study of gases and understand their behavior under various conditions. In the context of this exercise, we are dealing with an ideal gas undergoing reversible isothermal expansion, which means it is expanding at a constant temperature.
  • Ideal Gas Law: The behavior of ideal gases is described by the ideal gas law, \( PV = nRT \), where: \( P \) = pressure, \( V \) = volume, \( n \) = number of moles, \( R \) = gas constant, and \( T \) = temperature in Kelvin.
  • Assumptions: The particles in an ideal gas are assumed to have no volume and exert no forces on each other, except when they collide elastically.
Understanding these principles helps in solving problems related to the thermodynamic processes of ideal gases.
Entropy
Entropy is a measure of the disorder or randomness in a system. In thermodynamics, entropy quantifies the amount of energy in a system that is no longer available to do work. In the problem at hand, the entropy change is given, and you're asked to calculate the heat exchange.
  • Reversible Processes: In reversible processes, the system changes such that it can be returned to its original state without any net change in the system and surroundings.
  • Entropy Change Calculation: For reversible processes, changes in entropy \( \Delta S \) can be directly related to the heat transferred \( Q \) by the equation \( \Delta S = \frac{Q}{T} \).
Recognizing how entropy interacts with energy transfer forms the foundation for understanding various thermodynamic processes.
Heat Transfer
Heat transfer is the movement of thermal energy from one object or substance to another. In reversible isothermal expansion, we look at how much heat energy is transferred into or out of the gas as it expands.
  • Types of Heat Transfer: The main methods include conduction, convection, and radiation, but in thermodynamic problems, we often focus on the overall energy exchange represented by \( Q \).
  • Reversible Isothermal Expansion: During this expansion, the temperature remains constant and any heat input is used entirely to do work, leading to a direct link of heat with entropy through \( Q = T \Delta S \).
The comprehension of heat exchange during isothermal processes is crucial for mastering thermodynamics.
Thermodynamics
Thermodynamics is the branch of physics dealing with heat, work, and energy. It studies how energy is transferred in mechanical, chemical, and thermal forms. This exercise utilizes key thermodynamic concepts to understand the heat transfer in a reversible isothermal process.
  • First Law of Thermodynamics: It states that energy cannot be created or destroyed, only transformed. For a closed system, energy change \( \Delta U \) is given by the equation \( \Delta U = Q - W \), where \( W \) is the work done by the system.
  • Isothermal Process: In an isothermal process, the internal energy \( \Delta U \) remains unchanged and thus \( Q = W \).
Mastering thermodynamics equips you to analyze diverse physical processes from a fundamental energy perspective.

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Most popular questions from this chapter

Suppose \(4.00\) mol of an ideal gas undergoes a reversible isothermal expansion from volume \(V_{1}\) to volume \(V_{2}=2.00 V_{1}\) at temperature \(T=400 \mathrm{~K}\). Find (a) the work done by the gas and (b) the entropy change of the gas. (c) If the expansion is reversible and adiabatic instead of isothermal, what is the entropy change of the gas?

A \(10 \mathrm{~g}\) ice cube at \(-10^{\circ} \mathrm{C}\) is placed in a lake whose temperature is \(15^{\circ} \mathrm{C}\). Calculate the change in entropy of the cube-lake system as the ice cube comes to thermal equilibrium with the lake. The specific heat of ice is \(2220 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). (Hint: Will the ice cube affect the lake temperature?)

(a) During each cycle, a Carnot engine absorbs \(750 \mathrm{~J}\) as heat from a high-temperature reservoir at \(360 \mathrm{~K}\), with the low-temperature reservoir at \(280 \mathrm{~K}\). How much work is done per cycle? (b) The engine is then made to work in reverse to function as a Carnot refrigerator between those same two reservoirs. During each cycle, how much work is required to remove \(1200 \mathrm{~J}\) as heat from the lowtemperature reservoir?

The motor in a refrigerator has a power of \(200 \mathrm{~W}\). If the freezing compartment is at \(270 \mathrm{~K}\) and the outside air is at \(300 \mathrm{~K}\), and assuming the efficiency of a Carnot refrigerator, what is the maximum amount of energy that can be extracted as heat from the freezing compartment in \(10.0 \mathrm{~min} ?\)

A \(500 \mathrm{~W}\) Carnot engine operates between constant-temperature reservoirs at \(100^{\circ} \mathrm{C}\) and \(60.0^{\circ} \mathrm{C}\). What is the rate at which energy is (a) taken in by the engine as heat and (b) exhausted by the engine as heat?
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