Question

A heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross-section $A$ as shown in diagram. The cylinder contains one mole of an ideal diatomic gas. The current flows through the element such that the temperatures rises with time $t$ as $\Delta T=\alpha t+\frac{1}{2}\beta t^2$ ($\alpha$ and $\beta$ are constants), while pressure remains constant. The atmospheric pressure above the piston is $P_0$. Then:

• A. the rate of increase in internal energy is $\frac{5}{2}R(\alpha +\beta t)$
• B. the current flowing in the element is $\sqrt{\frac{5}{2r}R(\alpha +\beta t)}$
• C. the piston moves upwards with constant acceleration
• D. the piston moves upwards with constant speed

Answer 1

Answer: (A), (C)

the rate of increase in internal energy is $\frac{5}{2}R(\alpha +\beta t)$
the piston moves upwards with constant acceleration

For ideal diatomic gas, $C_v=\frac{5}{2}R$ and $C_p=\frac{7}{2}R$

Change in internal energy $\Delta U=n{C}_v\Delta T$ where $n=1$

Differentiating w.r.t time, $\frac{dU}{dt}=C_v\frac{dT}{dt}$

$\Delta T=\alpha t+\frac{1}{2}\beta t^2$ $⟹\frac{dT}{dt}=\alpha +\beta t$ ...............(1)

Thus rate of increase of internal energy $\frac{dU}{dt}=C_v(\alpha +\beta t)=$ $\frac{5}{2}R(\alpha +\beta t)$

Let the current flowing through the resistor be $i$

Using $Q=n{C}_p\Delta T$

Differentiating w.r.t time, $\frac{dQ}{dt}=C_p\frac{dT}{dt}$ where $dQ=i^{2}rdt$

$⟹$ $i^{2}r=C_p\frac{dT}{dt}$

$i^{2}r=\frac{7}{2}R(\alpha +\beta t)$ $⟹i=\sqrt{\frac{7R}{2r}(\alpha +\beta t)}$

Ideal gas equation : $pV=nRT$ $⟹\Delta V=\frac{R}{p}\Delta T$

$\therefore$ $\Delta x=\frac{R}{pA}\Delta T$ $(\because \Delta V=A\Delta x)$

Velocity of the piston $v=\frac{dx}{dt}=\frac{R}{pA}\frac{dT}{dt}=\frac{R}{pA}(\alpha +\beta t)$

Thus velocity of the piston depends on time and hence it is not constant.

Acceleration of the piston $a=\frac{dv}{dt}=\frac{R}{pA}\beta =constant$