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Chapter 10: Problem 31

The molar specific heats of an ideal gas at constant pressure and volume are denoted by \(C_{p}\) and \(C_{p}\), respectively. Further, \(\frac{C_{p}}{C_{v}}=\gamma\) and \(R\) is the gas constant for 1 gm mole of a gas. Then \(C_{v}\) is equal to (A) \(R\) (B) \(\gamma R\) (C) \(\frac{R}{\gamma-1}\) (D) \(\frac{\gamma R}{\gamma-1}\)

Short Answer

Expert verified
(C) \(\frac{R}{\gamma-1}\)

Step by step solution

01

Define given variables

The problem gives us the ratio of molar specific heats, defined as \(\gamma = \frac{C_{p}}{C_{v}}\). We also know that \(R\) is the gas constant.
02

Apply thermodynamics relationship

In thermodynamics, the specific heats at constant pressure and volume are related to the gas constant by the relation: \(C_p - C_v = R\). Let's use this relation to express \(C_v\) in terms of \(C_p\) and \(R\). So we have \(C_v = C_p - R\).
03

Substitute for \(C_p\)

For ideal gases, we can use the relation \(\gamma = \frac{C_{p}}{C_{v}}\), and express \(C_p\) as \(C_p = \gamma C_v\). Substitute this value into the previous equation, so we now have \(C_v = \gamma C_v - R\).
04

Solve the equation for \(C_v\)

Rearranging the equation results in: \((1 - \gamma) C_v = -R\) and hence, \(C_v = \frac{-R}{1-\gamma} = \frac{R}{\gamma-1}\). The denominator changes sign because \(\gamma > 1\) for gases. The negative sign thus needs to be removed to avoid a physical inconsistency.

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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 principle in chemistry and physics.
It simplifies the behavior of gases under different conditions of temperature and pressure. Expressed mathematically, the Ideal Gas Law is: \[ PV = nRT \] where:
  • \( P \) is the pressure of the gas
  • \( V \) is the volume of the gas
  • \( n \) is the number of moles
  • \( R \) is the gas constant
  • \( T \) is the temperature in Kelvin
This equation assumes that gas molecules do not interact with each other and that they occupy no volume themselves.
Therefore, it is an approximation useful for understanding real-world gases at standard conditions. The gas constant, \( R \), bridges the gap between individual molecules and bulk properties, allowing this equation to apply to any quantity of gas.
Specific Heat
Specific heat refers to the amount of heat required to change the temperature of a substance by one degree.
For gases, this is typically measured under two conditions: at constant volume and at constant pressure. For an ideal gas:- **Specific Heat at Constant Volume \( (C_v) \):**
Refers to the heat required to raise the temperature of one mole of gas by one degree while keeping the volume constant.- **Specific Heat at Constant Pressure \( (C_p) \):**
Refers to the same process but keeping the pressure constant instead.These values are crucial because they relate to how energy is stored or transferred within a system.
The relationship \( C_p - C_v = R \) links these specific heats to the gas constant, \( R \). This equation indicates how energy is absorbed by a gas when it expands.
Adiabatic Process
An adiabatic process describes a scenario where a gas evolves without exchanging heat with its surroundings.
This means any change in the gas's internal energy results from work done by or on the gas.For an adiabatic process, temperature changes, but heat transfer remains zero.
Adiabatic processes are characterized by the equation:\[ PV^\gamma = ext{constant} \]where:
  • \(P\) and \(V\) are the pressure and volume of the gas respectively
  • \(\gamma\) (Gamma) is the adiabatic index, defined as \(\frac{C_p}{C_v}\)
This exponent, \( \gamma \), indicates the pressure-volume relation during such processes
and differs from one type of gas to another. Understanding adiabatic processes is critical for analyzing engines and atmospheric dynamics.
Gas Constant
The gas constant, denoted by \( R \), is a key component in describing the state of a gas.
It appears in various gas laws like the Ideal Gas Law and plays a central role in thermodynamics. The value of \( R \) is approximately 8.314 J/(mol K).
It provides a link between macroscopic measurements like pressure and temperature
to the microscopic aspects of molecules. Moreover, \( R \) enables the calculation of specific heat capacities.
For instance, the difference in the specific heats, \( C_p - C_v = R \),
uses \( R \) to connect these thermodynamic properties.
Its significance lies in unifying diverse physical conditions into a single, universal expression.

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Most popular questions from this chapter

A constant volume gas thermometer shows pressure reading of \(50 \mathrm{~cm}\) and \(90 \mathrm{~cm}\) of mercury at \(0^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\), respectively. When the pressure reading is \(60 \mathrm{~cm}\) of mercury, the temperature is (A) \(25^{\circ} \mathrm{C}\) (B) \(40^{\circ} \mathrm{C}\) (C) \(15^{\circ} \mathrm{C}\) (D) \(12.5^{\circ} \mathrm{C}\)

1 mole of an ideal gas is contained in a cubical volume \(V\), ABCDEFGH at \(300 \mathrm{~K}\) (Fig. 10.18). One face of the cube (EFGH) is made up of a material which totally absorbs any gas molecule incident on it. At any given time, Fig. \(10.18\) (A) The pressure on \(E F G H\) would be zero. (B) The pressure on all the faces will be equal. (C) The pressure of \(E F G H\) would be double the pressure on \(A B C D\). (D) The pressure of \(E F G H\) would be half that on \(A B C D .\)

The average translational kinetic energy of 1 mole of \(O_{2}\) molecules (molar mass \(=32\) ) at a particular temperature is \(0.048 \mathrm{eV}\). The internal energy of 1 mole of \(N_{2}\) molecules (molar mass \(=28\) ) in \(\mathrm{eV}\) at same temperature is (A) \(0.048\) (B) \(0.003\) (C) \(0.0288\) (D) \(0.080\)

If \(\gamma\) be the ratio of specific heats of a perfect gas, the number of degrees of freedom of a molecule of the gas is (A) \((\gamma-1)\) (B) \(\frac{3 \gamma-1}{2 \gamma-1}\) (C) \(\frac{2}{\gamma-1}\) (D) \(\frac{9}{2}(\gamma-1)\)

Heat is associated with, (A) Kinetic energy of random motion of molecules. (B) Kinetic energy of orderly motion of molecules. (C) Total kinetic energy of random and orderly motion of molecules. (D) Kinetic energy of random motion in some cases and kinetic energy of orderly motion in other.
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