Work, heat and internal energy

Internal energy

Every system stores energy. Its molecules move, spin and vibrate. Its atoms are held together by bonds. Electrons and nuclei have energy too. The total of all this is the internal energy, $U$.

We cannot measure the total $U$ of a system. We do not need to. Chemistry cares about the change $\Delta U$ when the system goes from one state to another. That change we can measure.

$U$ changes in two ways: by work, and by heat. Matter going in or out would change it too, but in a closed system no matter crosses.

Changing U by work

Put some water in a thermos flask, so that no heat can get in or out. A change in which no heat crosses the boundary is adiabatic.

Now stir the water hard with a paddle. The water gets warmer, so its internal energy rises. The only thing that crossed the boundary was work, the work done by the paddle.

James Joule did experiments like this in the 1840s. He found something important. The same amount of work always gave the same change of state, whatever kind of work it was. Stirring, rubbing or passing a current all gave the same result.

So, for an adiabatic change:

This only makes sense if $U$ is a state function. The change depends on the start and the end, not on the kind of work.

Changing U by heat

Now put the same water in a beaker with thin metal walls. Stand it in a bath of hot water. Energy flows in through the walls, because the bath is hotter. This energy, which flows because of a difference in temperature, is heat, $q$.

If no work is done, the change in internal energy equals the heat:

Heat is not a substance in the water. It is energy on its way from a hotter body to a colder one. Once it has arrived, it is just part of $U$.

The sign convention

Chemistry uses one simple rule, as IUPAC recommends:

Energy that goes into the system is positive. Energy that leaves the system is negative.

• Heat absorbed by the system: $q > 0$.
• Heat given out by the system: $q < 0$.
• Work done on the system, for example compressing a gas: $w > 0$.
• Work done by the system, for example a gas pushing out a piston: $w < 0$.

Some physics books count work done by the system as positive. Always check which rule a book uses. In this chemistry course, we keep the rule above.

Heat and work together

Most changes involve both. Take a gas that absorbs 250 J of heat and does 100 J of work as it expands.

• $q = +250$ J, because heat goes in.
• $w = -100$ J, because the gas does work on the surroundings.
• The internal energy rises by $250 - 100 = 150$ J.

Here is another. A gas is compressed by 300 J of work, and gives out 120 J of heat. Then $w = +300$ J and $q = -120$ J. Its internal energy rises by $300 - 120 = 180$ J.

Set $q$ and $w$ to the second example. Then find two other pairs that give the same $\Delta U$. What does this tell you about $q$ and $w$ on their own?

Practise with new numbers

Try it in Python

The program works out $\Delta U$ for four systems from their heat and work. It also says whether $U$ rises, falls or stays the same.

Common mistakes

• Getting the sign of work wrong. A gas that expands does work on the surroundings, so $w$ is negative.
• Thinking an adiabatic change has no temperature change. It has no heat. Work can still change $U$ and the temperature.
• Saying a hot body contains heat. It contains internal energy. Heat is energy moving because of a temperature difference.
• Mixing up the conventions of two books. Decide on the sign rule before you start.

Remember these

• $U$ is a state function. We measure only $\Delta U$.
• Adiabatic: no heat crosses, so $\Delta U = w_\text{ad}$.
• No work: $\Delta U = q$.
• Energy into the system is positive, energy out is negative, for both $q$ and $w$.

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 is compressed inside a perfectly insulated cylinder with 90 J of work. What is $\Delta U$?

2. A gas gives out 75 J of heat while 40 J of work is done on it. What are $q$ and $w$?

3. For the gas in question 2, what is $\Delta U$?

4. Two students warm identical cups of water from the same start to the same end. One stirs hard. The other uses a hot plate. Is $\Delta U$ the same?