Question

When a gas expands adiabatically, it does work on its surroundings. But if there is no heat input to the gas, where does the energy come from to do the work?

Short answer

The energy to do work comes from the internal energy.

Step-by-step solution

Understanding the adiabatic process

A gas generally expands adiabatically means there is no heat transfer or exchange, that is Q = 0 (insulated). Work done can be measured in Joules.

Analysis of the reason behind the energy comes from to do the work

According to the first law of thermodynamics, the work done is given by:

$W=Q-∆U$

Here, Q = 0 and$∆U$ isthe change in internal energy.

The work done is given as:

$W=-∆U$

The work done is equal to the change in internal energy (where U decreases and$∆U$is negative) as the gas expands adiabatically.

So the internal energy (U) decreases, and this decrease in energy is utilized to do a work done. The temperature T will decrease as U decreases where$∆T\propto ∆U$ , according to the internal energy equation. It can be given as:

$∆U=n{C}_v∆T$

Here, n is the number of moles,$C_V$is the molar heat capacity at constant volume, and$∆T$is the change in the temperature.

Hence, the energy to do work comes from the internal energy.