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Chapter 8: Problem 1179

For adiabatic Process which relation is true mentioned below ? \(\gamma=\left\\{\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right\\}\) (A) \(\mathrm{p}^{\gamma} \mathrm{V}=\mathrm{Const}\) (B) \(\mathrm{T}^{\gamma} \mathrm{V}=\mathrm{Const}\) (C) TV \(^{\gamma}=\) Const (D) \(\mathrm{TV}^{\gamma-1}=\) Const

Short Answer

Expert verified
The correct relation for the adiabatic process is (D) \(TV^{γ-1} = Const\).

Step by step solution

01

Recognize the adiabatic process

In an adiabatic process, there is no heat transfer into or out of the system, i.e., the system is thermally isolated. The equation governing adiabatic processes for an ideal gas can be obtained by combining the ideal gas law and the definition of adiabatic processes.
02

Recall the ideal gas law

The ideal gas law is given by the equation: \(pV=nRT\), where p is pressure, V is volume, n is the number of moles of the gas, R is the gas constant, and T is temperature.
03

Derive the adiabatic process equation

For an adiabatic process, the relation between pressure, volume, and temperature can be derived from the ideal gas law and the adiabatic index (γ). By rearranging the ideal gas law and using the adiabatic index, we get the adiabatic process equation: \(pV^γ = C\), where C is a constant.
04

Compare the given options

Now let's compare the given options with the adiabatic process equation we derived: (A) \(p^γV = Const\) - This is not the correct relation. (B) \(T^γV = Const\) - This is not the correct relation. (C) \(TV^γ = Const\) - This is not the correct relation. (D) \(TV^{γ-1} = Const\) - This is the correct relation. By comparing the given options with the derived adiabatic process equation, we find that the correct relation for the adiabatic process is (D) \(TV^{γ-1} = Const\).

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Most popular questions from this chapter

The first law of thermodynamics is concerned with the conservation of (A) momentum (B) energy (C) mass (D) temperature

Starting with the same initial Conditions, an ideal gas expands from Volume \(\mathrm{V}_{1}\) to \(\mathrm{V}_{2}\) in three different ways. The Work done by the gas is \(\mathrm{W}_{1}\) if the process is purely isothermal, \(\mathrm{W}_{2}\) if purely isobaric and \(\mathrm{W}_{3}\) if purely adiabatic Then (A) \(\mathrm{W}_{2}>\mathrm{W}_{1}>\mathrm{W}_{3}\) (B) \(\mathrm{W}_{2}>\mathrm{W}_{3}>\mathrm{W}_{1}\) (C) \(\mathrm{W}_{1}>\mathrm{W}_{2}>\mathrm{W}_{3}\) (D) \(\mathrm{W}_{1}>\mathrm{W}_{3}>\mathrm{W}_{2}\)

What is an adiabatic Bulk modulus of hydrogen gas at NTP? (A) \(1.4\left(\mathrm{~N} / \mathrm{M}^{2}\right)\) (B) \(1.4 \times 10^{5}\left(\mathrm{~N} / \mathrm{M}^{2}\right)\) (C) \(1 \times 10^{-8}\left(\mathrm{~N} / \mathrm{M}^{2}\right)\) (D) \(1 \times 10^{5}\left(\mathrm{~N} / \mathrm{M}^{2}\right)\)

An ideal refrigerator has a freezer at a temperature of \(-13\) C, The coefficient of performance of the engine is 5 . The temperature of the air to which heat is rejected will be. (A) \(325^{\circ} \mathrm{C}\) (B) \(39^{\circ} \mathrm{C}\) (C) \(325 \mathrm{~K}\) (D) \(320^{\circ} \mathrm{C}\)

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) \(30^{\circ}\)
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