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Chapter 13: Problem 63

The slopes of isothermal and adiabatic curves are related as (a) isothermal curve slope \(=\) adiabatic curve slope (b) isothermal curve slope \(=\gamma \times\) adiabatic curve slope (c) adiabatic curve slope \(=\gamma \times\) isothermal curve slope (d) adiabatic curve slope \(=\frac{1}{2} \times\) isothermal curve slope

Short Answer

Expert verified
Option (c): adiabatic curve slope = \(\gamma \times\) isothermal curve slope.

Step by step solution

01

Understanding the Problem

We are given four options that describe the relationship between the slopes of isothermal and adiabatic curves for an ideal gas. We need to find which statement about their relationship is correct.
02

Recall Basic Concepts

If a process is isothermal, the temperature remains constant, and its equation is represented by the ideal gas law. In an adiabatic process, no heat is exchanged with the surroundings, and it follows the adiabatic condition. For an ideal gas, the adiabatic curve slope is steeper than the isothermal one.
03

Mathematical Representation of Slopes

For an isothermal process \(PV = constant\), the slope is \(-\frac{P}{V}\). For an adiabatic process \(PV^\gamma = constant\), the slope is \(-\gamma \frac{P}{V}\), where \(\gamma\) is the heat capacity ratio \(\frac{C_p}{C_v}\).
04

Comparison of Slopes

The comparison of slopes gives: the isothermal slope is \(-\frac{P}{V}\), while the adiabatic slope is steeper at \(-\gamma \frac{P}{V}\). Hence, the adiabatic curve's slope is \(\gamma\) times greater than the isothermal curve's slope.
05

Choose the Correct Option

From the description in the previous step, we know that the adiabatic curve slope equation includes \(\gamma\) multiplying the isothermal slope. Thus, the correct relationship between the slopes is option (c): adiabatic curve slope \(=\gamma \times\) isothermal curve slope.

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Key Concepts

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

Isothermal process
An isothermal process is a fundamental concept in thermodynamics where the temperature of a system remains constant throughout the process. This is typically an idealized process that can be achieved by ensuring that the system exchanges heat with its surroundings at exactly the right rate to maintain a steady temperature. The key equation governing this process is derived from the Ideal Gas Law.In an isothermal process:
  • The equation is given as: \( PV = ext{constant} \)
  • Here, \( P \) represents pressure and \( V \) represents volume.
  • Since temperature (\( T \)) is constant, any change in volume results in an opposite change in pressure.
  • The graph of an isothermal process is typically represented as a hyperbola on a Pressure-Volume (P-V) diagram.
This process is typically slow enough to allow the exchange of heat, ensuring temperature remains unaltered. In real-world applications, understanding isothermal processes helps in grasping concepts of engines and cooling systems.
Adiabatic process
An adiabatic process is another key concept in thermodynamics where the system does not exchange any heat with its surroundings. This isolation from heat transfer can occur if the process is carried out very quickly or if the system is well insulated.Key characteristics of an adiabatic process include:
  • The equation is given as: \( PV^\gamma = ext{constant} \)
  • Here, \( \gamma \) is the heat capacity ratio and differs for different gases.
  • Since no heat is transferred, changes in the internal energy of the system are entirely due to work done on or by the system.
  • An adiabatic process appears steeper than an isothermal process when plotted on a P-V diagram.
In practical terms, adiabatic processes are significant in understanding atmospheric phenomena and are also used to model the functioning of compression strokes in engines.
Ideal gas law
The Ideal Gas Law is a foundational principle that combines several simplified laws of gases. It provides a comprehensive description of the state of an ideal gas, relating its pressure, volume, and temperature.The law is expressed as:
  • \( PV = nRT \)
  • Where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature.
  • This equation is pivotal in connecting the physical properties of gases under various conditions.
  • It assumes that the gas behaves ideally, meaning it does not account for interactions between gas molecules or the volume occupied by them.
The Ideal Gas Law plays a crucial role across many scientific and engineering disciplines, from chemical reactions to designing systems like airbags where volume and pressure relationships are critical.
Heat capacity ratio
The Heat Capacity Ratio, also known as the adiabatic index or \( \gamma \), is an important concept in thermodynamics that helps determine how a gas behaves during adiabatic processes.Important aspects of heat capacity ratio:
  • It is defined as the ratio of the heat capacity at constant pressure \( C_p \) to the heat capacity at constant volume \( C_v \): \( \gamma = \frac{C_p}{C_v} \)
  • For ideal gases, \( \gamma \) is typically constant and greater than 1.
  • \( \gamma \) provides insight into the energy needed to change the temperature of a gas under constant pressure versus constant volume conditions.
  • It is crucial for assessing wave propagation speed in gases, influencing aspects like sound speed and shock waves.
Overall, understanding the heat capacity ratio is essential for analyzing thermodynamic processes, particularly those involving rapid pressure and volume changes.

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Most popular questions from this chapter

In an adiabatic process wherein pressure is increased by \(\frac{2}{3} \%\) if \(\frac{C_{p}}{C_{v}}=\frac{3}{2}\), then the volume decreases by about (a) \(\frac{4}{9} \%\) (b) \(\frac{2}{3} \%\) (c) \(4 \%\) (d) \(\frac{9}{4} \%\)

The maximum amount of heat that can be converted into mechanical energy, in any process (a) is \(100 \%\) (b) depends upon the temperatures at intake and exhaust (c) depends upon the amount of friction present (d) is the same for reversible and irreversible process

If for hydrogen \(C_{p}-C_{v}=m\) and for nitrogen \(C_{p}-C_{v}=\) \(n\), where \(C_{e}\) and \(C_{v}\) refer to specific heats per unit mass respectively at constant pressure and constant volume, the relation between \(m\) and \(n\) is (Molecular weight of hydrogen \(=2\) and molecular weight of nitrogen \(=14)\) (a) \(n=14 m\) (b) \(n=7 m\) (c) \(m=7 n\) (d) \(m=14 n\)

The work done in an adiabatic change in a particular gas depends upon only (a) change in volume (b) change in pressure (d) none of these (c) change in temperature

In a cyclic process, work done by the system is (a) zero (b) equal to heat given to the system (c) more than the heat given to the system (d) independent of heat given to the system
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