Question

$0.5$ moles of an ideal gas at constant temperature $27^{\circ} C$ kept inside a cylinder of length $L$ and cross-section area A closed by a massless piston. The cylinder is attached with a conducting rod of length $L$, cross-section area $\left(\frac{1}{9}\right) m ^{2}$ and thermal conductivity $k$, whose other end is maintained at $0^{\circ} C$. If piston is moved such that rate of heat flow through the conducing rod is constant then velocity of piston when it is at height $\frac{L}{2}$ from the bottom of cylinder is: [Neglect any kind of heat loss from system]

• A. $\left(\frac{ k }{ R }\right) m / \sec$
• B. $\left(\frac{ k }{10\, R }\right) m / \sec$
• C. $\left(\frac{ k }{ 100\,R }\right) m / \sec$
• D. $\left(\frac{ k }{ 1000\,R }\right) m / \sec$

Answer 1

Answer: (C)

$\frac{\Delta Q}{\Delta t}=\frac{\Delta W}{\Delta t}=$ work done per unit time $=\frac{k a \theta}{L}$
$\frac{d W}{d t}=P \frac{d v}{d t}=k \frac{a \theta}{L}, P=\frac{n R T}{V}$
$\Rightarrow \frac{0.5 R (300}{ V } A \cdot \frac{ d 1}{ dt }=\frac{ ka \theta}{ L }$
$\Rightarrow \frac{0.5 R (300)}{ A \cdot \frac{ L }{2}} A \cdot v =\frac{ ka \theta}{ L }$
$\Rightarrow v =\frac{ ka }{ R }\left(\frac{27}{300}\right)=\frac{ k }{100 R } .$