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Chapter 18: Problem 25

By what factor must the volume of a gas with \(\gamma=1.4\) be changed in an adiabatic process if the kelvin temperature is to double?

Short Answer

Expert verified
The factor by which the volume must be changed is \(\left(\frac{1}{2}\right)^{0.4}\)

Step by step solution

01

Understanding the Adiabatic Process

In an adiabatic process, there is no heat exchange with the surroundings. The equation that relates the volume (V), temperature (T) and the specific heat ratio (\(\gamma\)) of an ideal gas in this situation is given by: \[ V T^{(\gamma - 1)} = constant \]
02

Set Up the Equation

Let's assume the initial volume be \(V_1\) and the final volume be \(V_2\). Corresponding temperatures will be \(T_1\) and \(T_2\). We are given that the temperature is doubled \((T_2 = 2T_1)\) and \(\gamma = 1.4\). Using the equation from step 1, we can set up two equations:1. \( V_1 \cdot (T_1)^{(\gamma - 1)} = constant \)2. \( V_2 \cdot (T_2)^{(\gamma - 1)} = constant \)
03

Solve for the Unknown Factor

Since both equations equal the same constant, set them equal to each other:\( V_1 \cdot (T_1)^{(\gamma - 1)} = V_2 \cdot (T_2)^{(\gamma - 1)} \)Now, solve the equation above for \(V_2 / V_1\) (which indicates how much the volume must be changed):\[ \frac{V_2}{V_1} = \frac{T_1}{T_2}^{(\gamma - 1)} = \left(\frac{1}{2}\right)^{(1.4 - 1)} \]
04

Calculation

Substitute \(\frac{1}{2}\) and \(1.4 - 1 = 0.4\) into the equation to calculate the factor, resulting in: \( \frac{V_2}{V_1} = \left(\frac{1}{2}\right)^{0.4}\)

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