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Chapter 8: Problem 1178

What is the relationship Pressure and temperature for an ideal gas undergoing adiabatic Change. (A) \(\mathrm{PT}^{\gamma}=\) Const (B) \(\mathrm{PT}^{-1+\gamma}=\) Const (C) \(\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma}=\) Const (D) \(\mathrm{P}^{\gamma-1} \mathrm{~T}^{\gamma}=\) Const

Short Answer

Expert verified
The relationship between Pressure and temperature for an ideal gas undergoing adiabatic change is given by: \(\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma} = \text{Const}\) (Option C).

Step by step solution

01

Remember the adiabatic equation for an ideal gas

The adiabatic equation for an ideal gas is \(PV^{\gamma} = \text{constant}\), where P is pressure, V is volume, \(\gamma\) (gamma) is the specific heat ratio or adiabatic index (\(\frac{C_p}{C_v}\)), and \(PV^{\gamma}\) is constant during the adiabatic process.
02

Eliminate volume (V) term from adiabatic equation

To eliminate the volume (V) term, we can use the ideal gas equation, \(PV=nRT\), where n is the number of moles of the gas and R is the universal gas constant. Divide both sides by P: \(V = \frac{nRT}{P}\) Now substitute this expression for V in the adiabatic equation: \(P\left(\frac{nRT}{P}\right)^{\gamma} = \text{constant}\)
03

Simplify the equation

Distribute the \(\gamma\) exponent: \(P\left(\frac{n^{\gamma}R^{\gamma}T^{\gamma}}{P^{\gamma}}\right) = \text{constant}\) Simplify the equation by multiplying P: \(P^{1-\gamma}T^{\gamma} = \frac{n^{\gamma}R^{\gamma}}{\text{constant}}\) Since we need to find the relationship between P and T, both sides of the equation are constants, so it can be expressed as: \(P^{1-\gamma}T^{\gamma} = \text{constant}\)
04

Compare with given options

Based on our derived relationship between pressure and temperature for an ideal gas undergoing adiabatic change, the correct option is: (C) \(\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma} = \text{Const}\)

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Most popular questions from this chapter

The efficiency of heat engine is \(30 \%\) If it gives \(30 \mathrm{KJ}\) heat to the heat sink, than it should have absorbed ....... KJ heat from heat source. (A) \(42.8\) (B) 39 (C) 29 (D) 9

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A gas mixture consists of 2 mole of oxygen and 4 mole of argon at temperature \(\mathrm{T}\). Neglecting all vibrational modes, the total internal energy of the system is (A) \(11 \mathrm{RT}\) (B) \(9 \mathrm{RT}\) (C) \(15 \mathrm{RT}\) (D) \(4 \mathrm{RT}\)

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If a heat engine absorbs \(50 \mathrm{KJ}\) heat from a heat source and has efficiency of \(40 \%\), then the heat released by it in heat sink is (A) \(40 \mathrm{KJ}\) (B) \(30 \mathrm{KJ}\) (C) \(20 \mathrm{~J}\) (D) \(20 \mathrm{KJ}\)
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