All matter is made up of particles, which we will refer to here as ‘molecules’. Matter can have energy. For example, if we lift up a stone, it has gravitational potential energy. If we throw it, it has kinetic energy.
Kinetic and potential energies are the two general forms of energy. We consider the stone’s potential and kinetic energies to be properties or attributes of the stone itself; we calculate their values (mgh and ½mv2) using the mass and speed of the stone.
Now think about another way in which we could increase the energy of the stone: we could heat it (Figure 19.7). Now where does the energy from the heater go? The stone’s gravitational potential and kinetic energies do not increase; it is not higher or faster than before. The energy seems to have disappeared into the stone.

Of course, you already know the answer to this. The stone gets hotter, and that means that the molecules that make up the stone have more energy, both kinetic and electrical potential. They vibrate more and faster, and they move a little further apart. This energy of the molecules is known as the internal energy of the stone. The internal energy of a system (such as the heated stone) is defined as the sum of the random distribution of kinetic and potential energies of its atoms or molecules.

Molecular energy

Earlier in this chapter, where we studied the phases of matter, we saw how solids, liquids and gases could be characterised by differences in the arrangement, order and motion of their molecules. We could equally have said that, in the three phases, the molecules have different amounts of kinetic and potential energy.
Now, it is a simple problem to find the internal energy of an amount of matter. We add up the kinetic and potential energies associated with all the molecules in that matter. For example, consider the gas shown in Figure 19.8. There are ten molecules in the box, each having kinetic and potential energy. We can work out what all of these are and add them together, to get the total internal energy of the gas in the box.

Changing internal energy

There are two obvious ways in which we can increase the internal energy of some gas: we can heat it (Figure 19.9a), or we can do work on it by compressing it (Figure 19.9b).

Heating a gas

The walls of the container become hot and so its molecules vibrate more vigorously. The molecules of the cool gas strike the walls and bounce off faster. They have gained kinetic energy, and we say the temperature has risen.

Doing work on a gas

In this case, a wall of the container is being pushed inwards. The molecules of the cool gas strike a moving wall and bounce off faster. They have gained kinetic energy and again the temperature has risen.
This explains why a gas gets hotter when it is compressed.

Figure 19.9: Two ways to increase the internal energy of a gas: a by heating it, and b by compressing it.

There are other ways in which the internal energy of a system can be increased: by passing an electric current through it, for example. However, doing work and heating are all we need to consider here. 
The internal energy of a gas can also decrease; for example, if it loses heat to its surroundings, or if it expands so that it does work on its surroundings.

First law of thermodynamics 

You will be familiar with the idea that energy is conserved; that is, energy cannot simply disappear, or appear from nowhere. This means that, for example, all the energy put into a gas by heating it and by doing work on it must end up in the gas; it increases the internal energy of the gas. We can write this as an equation:
increase in internal energy = energy supplied by heating + work done on the system In symbols:

$\Delta U = q + W$

where $\Delta U$ is the increase in internal energy, q is the energy supplied to the system by heating and W is the work done on the system This is known as the first law of thermodynamics and is a formal statement of the principle of conservation of energy. (It applies to all situations, not simply to a mass of gas.) Since you have learned previously that energy is conserved, it may seem to be a simple idea, but it took scientists a good many decades to understand the nature of energy and to appreciate that it is conserved.
You should note the sign convention that is used in the first law. A positive value of $\Delta U$ means that the internal energy increases, a positive value of q means that heat is added to the system, and a positive value of W means that work is done on the system. Negative values mean that internal energy decreases, heat is taken away from the system or work is done by the system.
Imagine a gas heated from the outside in a sealed container of constant volume. In this case, no work is done on the gas as the heat is added, so W is 0 and the first law equation $\Delta U = q + W$ becomes $\Delta U = q$.
All the heat added becomes internal energy of the gas. If the container was able to expand a little as the heat is added then the situation needs some careful thought. Compressing a gas means work is done on the gas (W is positive); expanding a gas means work is done by the gas (W is negative as it pushes back and does work on the atmosphere). If the container expands then W is slightly negative and $\Delta U$ is slightly less than if the volume was constant.

A gas doing work

Gases exert pressure on the walls of their container. If a gas expands, the walls are pushed outwards – the gas has done work on its surroundings (W is negative, if the gas is the system). In a steam engine, expanding steam pushes a piston to turn the engine, and in a car engine, the exploding mixture of fuel and air does the same thing, so this is an important situation.

Figure 19.10 shows a gas at pressure p inside a cylinder of cross-sectional area A. The cylinder is closed by a moveable piston. The gas pushes the piston a distance s. If we know the force F exerted by the gas on the piston, we can deduce an expression for the amount of work done by the gas.
From the definition of pressure $(pressure = \frac{{force}}{{area}})$, the force exerted by the gas on the piston is given by:

$force = pressure \times area$
$F = p \times A$

and the work done is force × displacement:

$W = p \times A \times s$

But the quantity $A \times s$ is the increase in volume of the gas; that is, the shaded volume in Figure 19.10.
We call this $\Delta V$, where the $\Delta $ indicates that it is a change in V. Hence, the work done by the gas in expanding is:

$W = p\Delta V$

Notice that we are assuming that the pressure p does not change as the gas expands. This will be true if the gas is expanding against the pressure of the atmosphere, which changes only very slowly.
How does the first law of thermodynamics apply if you compress a gas? This can be done in different ways but we can consider two limiting ways.

Not allowing heat to enter or leave the system

This can be done by pushing the piston into the syringe very fast or by insulating the syringe. In this case, q is zero and $\Delta U = W$. All the work done by pushing in the piston increases the internal energy of the molecules. In this case, the kinetic energy of the molecules increases and the temperature increases, unless there is a change of state.

At constant temperature

Imagine pushing the piston very slowly into a syringe containing gas; so slowly that the temperature stays constant at room temperature. This change is known as an isothermal change. The kinetic energy of the molecules remains constant. 
The molecules become slightly closer together and this may mean that their internal energy U becomes slightly less but the change is very small (unless the gas becomes a liquid). If U is constant, then $\Delta U$ is zero and $0 = q + W$. This means that, if you push the piston in and do positive work W, then q is negative, and heat is lost from the syringe. You can think of this as doing positive work on the system and, with no extra internal energy, the system must lose some heat to the surroundings, perhaps by conduction of heat through the walls of the syringe. Similarly, if you pull the piston out very slowly, W is negative and q is positive and heat enters the system.