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Chapter 8: Problem 1094

Each molecule of a gas has \(\mathrm{f}\) degrees of freedom. The radio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (A) \(1+(\mathrm{f} / 2)\) (B) \(1+(1 / \mathrm{f})\) (C) \(1+(2 / \mathrm{f})\) (D) \(1+\\{(\mathrm{f}-1) / 3\\}\)

Short Answer

Expert verified
The correct option for the ratio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (C) \(1+(2 / \mathrm{f})\).

Step by step solution

01

Relationship between specific heat capacities and degrees of freedom

To find the relationship between the specific heat capacities and the degrees of freedom (\(f\)), we can recall that: \(C_p=C_v+nR\) Where \(C_p\) = specific heat capacity at constant pressure, \(C_v\) = specific heat capacity at constant volume, \(n\) = number of moles, and \(R\) = ideal gas constant.
02

Equation of specific heat capacity at constant volume

For a gas with \(f\) degrees of freedom, the specific heat capacity at constant volume (\(C_v\)) is given by: \(C_v=\frac{f}{2}nR\) We are going to use this equation to express \(\frac{C_p}{C_v}\) in terms of \(f\).
03

Derive the expression for gamma (\(\gamma\))

To find the ratio \(\gamma\), we can substitute the expression of \(C_v\) (from step 2) into the equation from step 1: \(C_p - C_v = nR\) Then divide both sides by \(C_v\): \(\frac{C_p}{C_v}-1 = \frac{2}{f}\) Add 1 on both sides of the equation to get the ratio: \(\frac{C_p}{C_v} = \gamma = 1 + \frac{2}{f}\) Comparing our derived expression with the given options, we find that it matches with option (C).
04

Answer

The correct option for the ratio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (C) \(1+(2 / \mathrm{f})\)

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Most popular questions from this chapter

For an adiabatic expansion of a perfect gas, the value of $\\{(\Delta \mathrm{P}) / \mathrm{P}\\}$ is equal to (A) \(-\sqrt{\gamma}\\{(\Delta \mathrm{r}) / \mathrm{v}\\}\) (B) \(-\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\) (C) \(-\gamma^{2}\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\) (D) \(-\gamma\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\)

Carnot engine working between \(300 \mathrm{~K}\) and \(600 \mathrm{~K}\) has work output of \(800 \mathrm{~J}\) per cycle. What is amount of heat energy supplied to the engine from source per cycle (A) \(1600\\{\mathrm{~J} /\) (cycle)\\} (B) \(2000\\{\mathrm{~J} /(\mathrm{cycle})\\}\) (C) \(1000\\{\mathrm{~J} /(\) cycle \()\\}\) (D) \(1800\\{\mathrm{~J} /(\) cycle \()\\}\)

Air is filled in a motor tube at \(27^{\circ} \mathrm{C}\) and at a Pressure of 8 atmosphere. The tube suddenly bursts. Then what is the temperature of air. given \(\gamma\) of air \(=1.5\) (A) \(150 \mathrm{~K}\) (B) \(150^{\circ} \mathrm{C}\) (C) \(75 \mathrm{~K}\) (D) \(27.5^{\circ} \mathrm{C}\)

A Centigrade and a Fahrenheit thermometer are dipped in boiling water. The water temperature is lowered until the Fahrenheit thermometer registered \(140^{\circ} .\) What is the fall in thermometers (A) \(80^{\circ}\) (B) \(60^{\circ}\) (C) \(40^{\circ}\) (D) \(30^{\circ}\)

Efficiency of a Carnot engine is \(50 \%\), when temperature of outlet is $500 \mathrm{~K}\(. in order to increase efficiency up to \)60 \%$ keeping temperature of intake the same what is temperature of out let. (A) \(200 \mathrm{~K}\) (B) \(400 \mathrm{~K}\) (C) \(600 \mathrm{~K}\) (D) \(800 \mathrm{~K}\)
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