question_answer

A 100 kg piston encloses 32 g of oxygen gas at a temperature of \[{{27}^{o}}C\] in a vertical cylinder of base area of \[4\text{ }d{{m}^{2}}\]. The air pressure outside is \[1\times {{10}^{5}}Pa\]. The axis of the cylinder is vertical, and the piston can move in it without friction. How much heat is to be transferred to the gas to raise the piston by 20 cm. Use \[R=\frac{25}{3}J/mol/K\]

A) 3500 J

B) 350 J

C) 7000 J

D) 750 J

Correct Answer: A

Solution :

[a] As heat is supplied to gas, it expands against atmospheric pressure and weight of piston. Thus the pressure of gas is constant given by,

\[P={{P}_{0}}+\frac{mg}{A}\]

\[P=1\times {{10}^{5}}+\frac{100\times 10}{4\times {{10}^{-2}}}=\left( 1+\frac{1}{4} \right){{10}^{5}}\]

\[=1.25\times {{10}^{5}}N/{{m}^{2}}\]

It is given that,

\[A=4d{{m}^{2}}=4\times {{10}^{-2}}{{m}^{2}}\]

\[m=100kg\]

\[n=\text{1mole}\]

\[V=\frac{nRT}{P}=2\times {{10}^{-2}}{{m}^{3}}\]

Initial height of piston from base of vessel,

\[h=\frac{V}{A}=\frac{2\times {{10}^{-2}}}{4\times {{10}^{-2}}}=\frac{1}{2}m\]

As process is isobaric, so

\[\frac{{{T}_{1}}}{{{T}_{2}}}=\frac{{{V}_{1}}}{{{V}_{2}}}=\frac{V}{V+A\Delta h}=\frac{h}{h+\Delta h}\]

\[\therefore \]    \[\frac{{{T}_{2}}-{{T}_{1}}}{{{T}_{1}}}=\frac{\Delta h}{h}\]

\[\Delta Q=n{{C}_{p}}({{T}_{2}}-{{T}_{1}})=\frac{n.{{C}_{p}}.{{T}_{1}}}{h}\Delta h\]

\[=\frac{7}{2}\times \frac{25}{3}\times \frac{300}{0.5}\times 0.2=350\text{0 J}\]