Question: (II) Use the conservation of energy to explain why the temperature of a well-insulated gas increases when it is compressed—say, by pushing down on a piston—whereas the temperature decreases when the gas expands. Show your reasoning.

The first law of thermodynamics says the entire energy of the universe is always conserved. The variation in internal energy of the system is related to the heat transfer to or from the system and the work done on or by the system.

Mathematically, it can be expressed as,

\(\Delta U = Q - W\) … (i)

Here, Q is the heat transfer, W is the work done, and \(\Delta U\) is the change in internal energy.

The gas is well insulated that means there is no heat transfer between system and surroundings. That is\(Q = 0\).

So, the equation (i) can be written as:

\(\Delta U = - W\) … (ii)

During the compression process, the volume of the gas reduces. That indicates the work is done on the system. By convention, work done on the system is negative. By plugging the negative value of work done in the above equation (ii), the change in the internal energy of the system becomes positive. It shows that there is an improvement in the internal energy of the system during compression. An increase in internal energy shows that the temperature of the system increases.

During the expansion process, the gas is doing work on the surrounding. By convention, work done by the system is positive. By substituting positive work into equation (ii), it indicates that the variation in internal energy becomes negative. That suggests there is a reduction in the system's internal energy, which indicates that the temperature of the system goes down during the expansion of the gas.