The adiabatic process

No heat in or out

If a gas is well insulated, or changed so quickly that heat has no time to flow, $Q = 0$. Then the first law gives

Work done on the gas raises its temperature; work done by it lowers its temperature. Pump a bicycle tyre hard and the pump warms; let a can of spray out fast and it cools.

The adiabatic laws

For an ideal gas changing slowly enough to stay in balance but with no heat flow:

and the work done by it is $W = nC_v(T_1 - T_2) = \dfrac{p_1V_1 - p_2V_2}{\gamma - 1}$.

You should find that the adiabatic pressure falls faster, because the gas also cools, and that the gap grows with $\gamma$.

Watch out. $pV^\gamma$, not $pV$: Boyle's law does not hold when the temperature changes.

A worked example

A diesel engine squeezes air ($\gamma = 1.40$) at $300\ \text{K}$ and $1.0\ \text{atm}$ quickly to $\tfrac{1}{18}$ of its volume. Find the new temperature and pressure.

1. Temperature. $T_2 = 300\ \text{K} \times 18^{0.40} = 953\ \text{K}$, about $680\ ^\circ\text{C}$.
2. Pressure. $p_2 = 1.0\ \text{atm} \times 18^{1.40} = 57\ \text{atm}$.
3. So the air is hot enough to light the fuel sprayed into it, with no spark plug.

Practise with new numbers

Try it in Python

The program works out the diesel engine of the example.

Remember these

• Adiabatic: $Q = 0$, $W = -\Delta U = nC_v(T_1 - T_2)$.
• $pV^\gamma$ and $TV^{\gamma - 1}$ are constant.
• Squeezed: warms. Expands: cools. Adiabats are steeper than isotherms.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A monatomic gas expands adiabatically to $8$ times its volume. By what factor does its temperature fall?

2. In an adiabatic expansion, a gas does $600\ \text{J}$ of work. What is $\Delta U$?

3. Why does a bicycle pump get hot?

4. Why are clouds made when moist air rises?