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

Answer 1

Answer: (C)

$\frac{\Delta Q}{\Delta t}=\frac{\Delta W}{\Delta t}=$ work done per unit time $=\frac{ka\theta }{L}$
$\frac{dW}{dt}=P\frac{dv}{dt}=k\frac{a\theta }{L},P=\frac{nRT}{V}$
$\Rightarrow \frac{0.5R(300}{V}A\cdot \frac{d1}{dt}=\frac{ka\theta }{L}$
$\Rightarrow \frac{0.5R(300)}{A\cdot \frac{L}{2}}A\cdot v=\frac{ka\theta }{L}$
$\Rightarrow v=\frac{ka}{R}\left(\frac{27}{300}\right)=\frac{k}{100R}.$