A vertical cylinder contains v moles of an ideal gas and is closed off by a piston of mass M and area A. The acceleration due to gravity is g. The molar specific heat cv (at constant volume) of the gas is a constant independent of temperature. The heat capacities of the piston and cylinder are negligibly small and any frictional forces between the piston and the cylinder walls can be neglected. The whole system is thermally insulated. Initially, the piston is clambed in position so that the gas has a volume Vo and a temperature To- The piston is now released and, after some oscillations, comes to rest in a final equilibrium situation corresponding to a larger volume of the gas. Note: This is not a quasistatic process, though it is adiabatic. You may assume that the force due to atmospheric pressure on the piston is negligible compared to Mg. You should be able to answer parts (a) and (b) just from general considerations, without any calculation. (a) Does the temperature of the gas increase, decrease, or remain the same? (b) Does the entropy of the gas increase, decrease, or remain the same? (c) Calculate the final temperature of the gas in terms of To, Vo, and the other parameters mentioned in the statement of the problem. [Conservation of energy is key here.]

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$(a)$ According to the first law of thermodynamics,

$∆ U = Q + W$

Here, $∆ U$ is the change in internal energy, $Q$ is heat added, and $W$ is work done on the system.

System is thermally isolated. Therefore,

$Q = 0$

The expanding gas does work on the surroundings is, $W < 0$

Thus, the first law of thermodynamics gives,

$∆ U < 0$

For an ideal gas, the internal energy $U$ is a function of temperature $T$ only, and hence the gas cools.

Hence, as the gas expands, the work is done by the gas at the cost of its internal energy. So the temperature decreases.

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