Thermodynamic Systems, Internal Energy and the Work Done by a Gas

1A lid that dances

Put a pan of water on the fire with its lid on. As the water boils, the lid begins to rattle and lift: steam pushes it up, again and again. Heat from the fire has become motion of the lid. This simple kitchen sight holds the idea behind every steam engine, petrol engine and power station: a hot gas doing work. Thermodynamics is the physics of how heat, work and the energy stored in matter are related. In S2 you met the ideal gas laws, and in S3 heat capacity and heat transfer. In this unit you combine them into the laws of thermodynamics, starting here with the language of systems, internal energy and the work done by a gas.

2Systems and their surroundings

A thermodynamic system is the part of the universe we choose to study: the gas in a cylinder, the water in a kettle, a whole engine. Everything outside it is the surroundings, and the two are separated by a boundary, real or imaginary. Systems are classified by what can cross the boundary:

Table 1.1: Open, closed and isolated systems.

The condition of a system in equilibrium is described by a few state variables: pressure $pp$, volume $VV$, temperature $TT$ and the amount of substance $nn$. For an ideal gas they are linked by the equation of state you met in S2, $pV=nRTpV\; =\; nRT$, with $R=8.31\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}R\; =\; 8.31\backslash ,\backslash mathrm\{J\backslash ,mol^\{-1\}\backslash ,K^\{-1\}\}$.

3Internal energy, heat and work

The molecules of any substance move randomly and pull or push on one another. The internal energy $UU$ of a system is the total of the random kinetic energies and the potential energies of all its molecules. In an ideal gas the molecules exert no forces except in collisions, so $UU$ is all kinetic energy, and it depends only on temperature. For a monatomic ideal gas such as helium or argon,

Internal energy is a function of state: it depends only on the present state of the system, not on how it got there. There are two ways to change it:

• heating $QQ$: energy transferred because of a temperature difference between the system and its surroundings;
• work $WW$: energy transferred when a force moves its point of application, for example when a gas pushes a piston.

Heat and work are energy in transit. A gas does not "contain" heat or work; it contains internal energy.

4Work done by an expanding gas

Gas at pressure $pp$ in a cylinder pushes on a piston of area $AA$ with force $F=pAF\; =\; pA$ (Figure 1.1). If the piston moves out a small distance $\mathrm{\Delta }x\backslash Delta\; x$, keeping the pressure nearly constant, the work done by the gas is

When the gas expands ($\mathrm{\Delta }V>0\backslash Delta\; V\; \backslash gt\; 0$) it does positive work on the surroundings; when it is compressed ($\mathrm{\Delta }V<0\backslash Delta\; V\; \backslash lt\; 0$) the work done by it is negative, which means work is done on it. If the pressure changes during the expansion, add up the small pieces: $W=\int p\mathrm{d}VW\; =\; \backslash int\; p\backslash ,\backslash mathrm\{d\}V$, which is the area under the curve on a p–V graph.

Figure 1.1: A gas expanding against a piston does work W = pΔV.

5Different paths between the same two states

A gas can be taken from state A to state B in many ways. In Figure 1.2 the gas starts at A (1.0 L, $2.0\times {10}^5\mathrm{P}\mathrm{a}2.0\; \backslash times\; 10^\{5\}\backslash ,\backslash mathrm\{Pa\}$) and ends at B (3.0 L, $1.0\times {10}^5\mathrm{P}\mathrm{a}1.0\; \backslash times\; 10^\{5\}\backslash ,\backslash mathrm\{Pa\}$). It can expand at high pressure and then cool at constant volume (path ACB), cool first and then expand at low pressure (ADB), or change both together (the straight line AB). The areas under the three paths differ, so the work done depends on the path, and so (as you will see in Lesson 2) does the heat. But $pVpV$ is the same at A and B ($200\mathrm{J}200\backslash ,\backslash mathrm\{J\}$ in each case), so the temperature, and the internal energy, are the same at both ends whatever the path.

Figure 1.2: Three paths from A to B on a p–V diagram, with the work done along each.

A change that returns the gas to its starting state is a cycle. Going round a cycle, $\mathrm{\Delta }U=0\backslash Delta\; U\; =\; 0$, and the net work done by the gas is the area enclosed by the loop: positive if the loop is traced clockwise. This is how an engine turns heat into work over and over again.