Question

Find the final temperature of one mole of an ideal gas that has an initial temperature $tK$. The gas does $9R$ joules of work adiabatically. The ratio of specific heats of this gas at constant pressure and at constant volume is $\frac{4}{3}$.

Answer 1

Step 1: Given from the question

Given, $T_{initial}=tK$

Work, $W=9R$

Ratio of specific heats

$\gamma =\frac{C_p}{C_v}=\frac{4}{3}$

Step 2: Calculate the Final Temperature

In an adiabatic process, work done is given as,

$W=\frac{R\left(T_{final}-T_{initial}\right)}{1-\gamma }$

$\Rightarrow$$9R=\frac{R\left(T_{final}-t\right)}{1-{\displaystyle \frac{4}{3}}}$

$\Rightarrow$$9\left(1-\frac{4}{3}\right)=T_{final}-t$

$\Rightarrow$ $T_{final}=t-\frac{9}{3}$

$=t-3$

Hence, the required final temperature is $t-3$.