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 tha the temp rature rises with time t as $△T=\alpha t+\frac{1}{2}\beta t^2$ ($\alpha$ 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 correct options are
A The rate of increase in internal energy is $\frac{5}{2}R(\alpha +\beta t)$
C The piston moves upwards with constant acceleration

Internal energy, $U=\frac{nfRT}{2}=\frac{5R}{2}t\frac{1}{2}{t}^2$

$⟹\frac{dU}{dt}=\frac{5}{2}R(\alpha +\beta t)$

$dQ=n{C}_{P}dT⟹\frac{dQ}{dT}=n{C}_P\frac{dT}{dt}⟹i^{2}r=\frac{7}{2}Rt⟹i=\sqrt{\frac{5}{2r}R(\alpha +\beta t)}$

$PV=nRT⟹V=\frac{nRT}{P}⟹V=\frac{nR}{P}t\frac{1}{2}{t}^2⟹x=\frac{nR}{PA}t\frac{1}{2}{t}^2⟹v=\frac{nR}{PA}t$

This gives accelearation=$\frac{nR}{PA}$