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Chapter 18: Problem 37

A gas undergoes an adiabatic compression during which its volume drops to half its original value. If the gas pressure increases by a factor of \(2.55,\) what's its specific-heat ratio \(\gamma ?\)

Short Answer

Expert verified
The specific heat ratio \(\gamma\) is approximately 1.322

Step by step solution

01

Express the Given Parameters in Terms of the Adiabatic Law

First, express the adiabatic law in terms of the given parameters: \(\frac{P_1}{P_2} = \left(\frac{V_2}{V_1}\right)^\gamma\). From the problem, we know that \(P_2 = 2.55P_1\) and \(V_2 = 0.5V_1\). So we can substitute these values into the equation: \(\frac{1}{2.55} = \left(\frac{0.5}{1}\right)^\gamma\)
02

Solve for \(\gamma\)

Now, we isolate \(\gamma\) by taking the logarithm base 0.5 on both sides of the equation: \(\log_{0.5}\left(\frac{1}{2.55}\right) = \gamma\)
03

Evaluate \(\gamma\)

Finally, evaluate the logarithm on the left side to get your final answer for \(\gamma\). This can be done using a scientific calculator or a computer program that can perform logarithmic calculations.

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Most popular questions from this chapter

A 0.25 -mol sample of ideal gas initially occupies \(3.5 \mathrm{L}\). If it takes 61 J of work to compress the gas isothermally to 3.0 L, what's the temperature?

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