An ideal gas is compressed adiabatically from volume $V$ to $V/8$. If the initial temperature is $300\text{ K}$ and the adiabatic index is $1.5$, what is the final temperature?

Solution

1. Given the initial temperature $T_1=300\text{ K}$, initial volume $V_1=V$, final volume $V_2=V/8$, and adiabatic index $\gamma =1.5$.
2. Use the adiabatic relation $T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$ to relate the initial and final states.
3. Substitute the given values into the equation: $300\cdot V^{1.5-1}=T_2\cdot (V/8{)}^{1.5-1}$.
4. Simplify the expression to $300\cdot V^{0.5}=T_2\cdot (V/8{)}^{0.5}$, which yields $$\begin{array}{rl}{T}_2& =300\cdot 8^{0.5}\\ & =300\cdot \sqrt{8}\approx 300\cdot 2.828\end{array}$$ is incorrect; note that $$\begin{array}{rl}{8}^{0.5}& =\sqrt{8}\\ & =2\sqrt{2}\end{array}$$ is not the intended path, rather $$\begin{array}{rl}{T}_2& =300\cdot (V/(V/8){)}^{0.5}\\ & =300\cdot 8^{0.5}\\ & =300\cdot 4\\ & =1200\text{ K}\end{array}$$.
5. Hence the answer is (C).