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Chapter 12: Problem 2

For an adiabatic expansion process, the quantity \(P V\) (A) decreases. (B) increases. (C) remains constant. (D) depends on adiabatic exponent of the gas.

Short Answer

Expert verified
The behavior of \(PV\) during an adiabatic expansion process depends on the adiabatic exponent \(\gamma\), as indicated by the adiabatic equation \(PV^{\gamma} = constant\). Therefore, the correct answer is (D) depends on the adiabatic exponent of the gas.

Step by step solution

01

Review the adiabatic equation

For adiabatic processes, we will use the following equation, which relates pressure, volume, and the adiabatic exponent, denoted as \(\gamma\): \(PV^{\gamma} = constant\) Here, \(P\) represents pressure \(V\) represents volume \(\gamma\) represents the adiabatic exponent
02

Analyze the adiabatic expansion process

In an adiabatic expansion process, the volume of the gas increases. Since the left side of the adiabatic equation must always remain constant, we need to observe how the product PV will change as the volume increases.
03

Determine the behavior of PV during adiabatic expansion

Let's differentiate the adiabatic equation with respect to volume and set it equal to zero, as the left-hand side is a constant: \(\frac{d}{dV}(PV^{\gamma}) = 0\) Using the product rule and chain rule: \(P\frac{d}{dV}(V^{\gamma})+\gamma V^{\gamma}\frac{dP}{dV} = 0\) Since we are considering an adiabatic expansion, the volume (\(V\)) increases, which means \(dV\) is positive. Now, we need to analyze the behavior of \(PV\). If the product \(PV\) remains constant, then the term \(\frac{dP}{dV}\) will be zero. However, looking at our differentiated equation, we see that this is not the case.
04

Choosing the correct answer

Given our analysis, we can conclude that the product \(PV\) does not remain constant during an adiabatic expansion process (eliminating option C). At this point, we don't have enough information to definitively choose between options A, B, or D, but we can see that the behavior of \(PV\) depends on the adiabatic exponent \(\gamma\) and the specific process of the expansion (A and B are not always true). Therefore, the correct answer is: (D) depends on the adiabatic exponent of the gas.

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Most popular questions from this chapter

Temperature of \(100 \mathrm{gm}\) water in changed from \(0^{\circ} \mathrm{C}\) to \(3^{\circ} \mathrm{C}\). In this process, heat supplied to water will be (specified heat of water \(=1 \mathrm{cal} / \mathrm{gm}{ }^{\circ} \mathrm{C}\) ) (A) Equal to \(300 \mathrm{cal}\) (B) Greater than \(300 \mathrm{cal}\) (C) Less than \(300 \mathrm{cal}\) (D) Data is insufficient

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A vertical cylinder with a massless piston is filled with one mole of an ideal gas. The piston can move freely without friction. The piston is slowly raised so that the gas expands isothermally at temperature \(300 \mathrm{~K}\). The amount of work done in increasing the volume two times is \(R=\frac{25}{3} \mathrm{~J} / \mathrm{mol} / \mathrm{K}, \log _{\mathrm{e}} 2=0.7\) (A) \(1750 \mathrm{~J}\) (B) \(2500 \mathrm{~J}\) (C) \(750 \mathrm{~J}\) (D) \(4250 \mathrm{~J}\)

When a system is taken from state \(i\) to state \(f\) along the path iaf, it is found \(Q=50 \mathrm{cal}\) and \(W=20\) cal. Along the path \(i b f, Q=36\) cal. \(W\) along the path \(i b f\) is [2007] (A) \(6 \mathrm{cal}\) (B) \(16 \mathrm{cal}\) (C) \(66 \mathrm{cal}\) (D) \(14 \mathrm{cal}\)
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