Question

A sample of air is kept in a container having walls which are slightly conducting. The initial temperature and volume are ${27}^{\circ }\text{C}$ and $800{\text{cm}}^3$ respectively. Find the rise in temperature if the gas is compressed to $200{\text{cm}}^3$ $\left(i\right)$ in a short time $\left(ii\right)$ in a sufficiently long time.
[Take $\gamma =1.4$ and $4^{0.4}=1.74$]

• A. $\left(i\right)522\text{K}$ $\left(ii\right)0\text{K}$
• B. $\left(i\right)0\text{K}$ $\left(ii\right)522\text{K}$
• C. $\left(i\right)222\text{K}$ $\left(ii\right)0\text{K}$
• D. $\left(i\right)0\text{K}$ $\left(ii\right)222\text{K}$

Answer 1

The correct option is C $\left(i\right)222\text{K}$ $\left(ii\right)0\text{K}$

When the gas is compressed in a short interval of time, the process can be considered to be abiabatic. Thus, the equation of state in terms of $T$ and $V$ gives:
$T_2{V}_2^{\gamma -1}=T_1{V}_1^{\gamma -1}$
or $T_2=T_1{\left(\frac{V_1}{V_2}\right)}^{\gamma -1}$
From the data given in the question,
$T_2=300\times {\left[\frac{800}{200}\right]}^{0.4}=522\text{K}$
$\therefore$ Rise in temperature $\Delta T=T_2-T_1=222\text{K}$

When the gas is compressed in a sufficiently long time, the process can be considered to be isothermal. Thus, the temperature remains constant i.e ${27}^{\circ }\text{C}.$
So, rise in temperature $\Delta T=0$ .
Thus, option (c) is the correct answer.