The first law of thermodynamics

Energy is conserved

A gas in a cylinder can get energy as heat from a flame, and lose energy by pushing a piston out. Whatever is left over changes its internal energy:

This is the first law of thermodynamics, the conservation of energy written for heat and work.

The signs

• $Q > 0$: heat goes into the gas; $Q < 0$: heat comes out.
• $W > 0$: the gas does work (it expands); $W < 0$: work is done on it (it is squeezed).

You should find that with no heat, the work comes straight out of the internal energy: an insulated gas that expands cools.

Special cases

• Insulated ($Q = 0$): $\Delta U = -W$.
• Rigid container ($W = 0$): $\Delta U = Q$.
• A cycle (back to the start): $\Delta U = 0$, so $Q = W$.
• Ideal gas at a steady temperature: $\Delta U = 0$, so $Q = W$.

State and path

$U$ depends only on the state: going from A to B by any route, $\Delta U$ is the same. $Q$ and $W$ depend on the route, but their difference does not.

Joule showed that work and heat are the same thing in two forms: falling weights turning paddles in water warm it exactly as much as the matching amount of heat.

Watch out. Keep the signs. 'Work done on the gas' is negative $W$ in this convention.

A worked example

A gas is heated with $800\ \text{J}$ and pushes a piston out, doing $300\ \text{J}$ of work. Later it is squeezed by $390\ \text{J}$ of work while it loses $170\ \text{J}$ of heat. Find $\Delta U$ each time.

1. First. $\Delta U = 800\ \text{J} - 300\ \text{J} = +500\ \text{J}$.
2. Second. $Q = -170$, $W = -390$: $\Delta U = -170\ \text{J} + 390\ \text{J} = +220\ \text{J}$.
3. So both raise its internal energy, though in the second case heat leaves it.

Practise with new numbers

Try it in Python

The program applies the first law to four cases.

Remember these

• $\Delta U = Q - W$; $Q$ in is positive, $W$ by the gas is positive.
• Insulated: $\Delta U = -W$. Rigid: $\Delta U = Q$. Cycle: $Q = W$.
• $U$ is a state function; $Q$ and $W$ depend on the path.

Check yourself

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

1. A gas takes in $1200\ \text{J}$ and does $700\ \text{J}$ of work. What is $\Delta U$?

2. In a cycle, a gas does $350\ \text{J}$ of net work. How much net heat did it take in?

3. Why does air let out of a tyre feel cold?

4. Is the first law broken by a machine that gives out more work than the energy put in?