Question
A cylinder with thermally insulated walls contains an insulated portion which can slide freely. The partition divides the cylinder into two chambers containing equal moles of the same gas, the initial pressure, temperature and volume being Po, To and Vo, respectively. By means of heating the coil, heat is supplied slowly to the gas in one chamber (A) until its pressure becomes 27Po/8. If the value of $\gamma$ is 1.5, then find the heat supplied to the gas in chamber A.
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Solution
Chamber B: $P_{0}.V_{0}^{\gamma} = \frac{27P_{0}}{8}.V_{B}^{\gamma}$
$\therefore V_{B} = \frac{4V_{0}}{9} \Rightarrow T_{B} = \frac{3}{2}T_{0} $$${\text{and }V_{A} = 2V_{0} - \frac{4V_{0}}{9} = \frac{14V_{0}}{9} \Rightarrow T_{A} = \frac{21}{4}T_{0} }{\text{Now, }q_{A} = \Delta U_{A} + \Delta U_{B} = n.C_{V,m}.\left( T_{A} - T_{0} \right) + n.C_{V,m}.\left( T_{B} - T_{0} \right) }{= \frac{P_{0}V_{0}}{R.T_{0}}.2R.\left\lbrack \left( \frac{21}{4}T_{0} - T_{0} \right) + \left( \frac{3}{2}T_{0} - T_{0} \right) \right\rbrack = \frac{19}{2}.P_{o}V_{0}}$$
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