The initial pressure and volume of a given mass of an ideal gas $\left(with\frac{C_{p}}{C_{v}} =\gamma\right),$ taken in a cylinder fitted with a piston, are P$ _0$ and V$_0$ respectively. At this stage the gas has the same temperature as that of the surrounding medium which is T$_0$. It is adiabatically compressed to a volume equal to $\frac{v_{0}}{2}.$ Subsequently the gas is allowed to come to thermal equilibrium with the surroundings. What is the heat released to the surroundings ?
- 0
- $\left(2^{\gamma-1}-1\right)\frac{P_{0} V_{0}}{\gamma-1}$
- $\gamma P_{0}V_{0} in 2$
- $\frac{P_{0}V_{0}}{2\left(\gamma-1\right)}$
The Correct Option is B
Solution and Explanation
$\therefore After compression, we assume the piston to be fixed.$
$\therefore\Delta Q =nC_{v}\Delta T =n\frac{R}{\gamma-1}\left(T_{0}-T_{0}2^{\gamma-1}\right) =\frac{P_{0}V_{0}}{\gamma-1}\left(1-2^{\gamma-1}\right)$
$\therefore heat released =\frac{P_{0}V_{0}}{\gamma-1} \left(2^{\gamma-1}-1\right)$
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Concepts Used:
Work Done Thermodynamics
In thermodynamics, work is a way of energy transfer from a system to surroundings, under the influence of external factors such gravity, electromagnetic forces, pressure/volume etc.
Energy (ΔU) can cross the boundary of a system in two forms -> Work (W) and Heat (q). Both work and heat refer to processes by which energy is transferred to or from a substance.
ΔU=W+q
Work done by a system is defined as the quantity of energy exchanged between a system and its surroundings. It is governed by external factors such as an external force, pressure or volume or change in temperature etc.
Work (W) in mechanics is displacement (d) against a resisting force (F).
Work has units of energy (Joule, J)