Internal Energy

Every object contains energy stored in its molecules: the energy of their random jiggling and the energy locked in the forces between them. This is internal energy. It explains why heating raises temperature, why melting ice stays at $0\ ^\circ\text{C}$, and why the internal energy of an ideal gas depends only on its temperature. The definition is examined word for word in Paper 4, and questions on it are often the first part of a longer first law of thermodynamics question.

What internal energy is

Zoom in on a block of copper. Its atoms are not still: each one vibrates about its position in the lattice, with kinetic energy that changes from moment to moment. The atoms are also held in place by electrical forces from their neighbours, so each has potential energy depending on how far it is from its neighbours. Neither energy is the same for every atom; at any instant some atoms are moving fast, some slowly, some are squeezed close, some stretched apart. The energies are randomly distributed.

Add all of these up and you get the internal energy.

Each phrase matters:

• Sum: the total over all the molecules, not an average.
• Random distribution: the energies are shared out randomly among the molecules. This excludes ordered energy such as the kinetic energy of the block flying through the air, or its gravitational potential energy on a shelf. Throwing a block does not change its internal energy.
• Kinetic energy: from the random translational, rotational and vibrational motion of the molecules.
• Potential energy: from the intermolecular forces, which depends on the separation of the molecules.

The two parts in solids, liquids and gases

State	Molecular kinetic energy	Molecular potential energy
Solid	Vibration about fixed positions	Large and negative: strong bonds, molecules close together
Liquid	Vibration and slow movement past each other	Still large and negative, slightly less so than in the solid
Gas	Fast random translation (and rotation)	Close to zero: molecules far apart, forces tiny
Ideal gas	All of the internal energy	Exactly zero: no intermolecular forces

The potential energy of bonded molecules is negative, for the same reason that gravitational potential energy is negative: zero is defined as infinite separation, and the molecules attract, so energy must be supplied to pull them apart to zero. Breaking bonds (melting, boiling) raises the potential energy towards zero.

Internal energy is a function of state

The state of a system is described by its measurable properties: for a gas, its pressure, volume and temperature (and the amount present). Internal energy is determined by the state of the system. If you know the state, you know the internal energy, however the system got there.

An analogy: your height above sea level depends only on where you are standing, not on the path you climbed. Internal energy depends only on the state, not on whether it was reached by heating, by doing work, or by some combination.

This has two practical consequences:

• If a system goes through a series of changes and returns to its starting state (a cycle), its internal energy returns to its starting value. The total change in internal energy around any cycle is zero.
• Different processes between the same two states always give the same $\Delta U$, even though the heating and work done may be completely different.

The amounts of heating and work done are not functions of state. A system does not "contain heat" or "contain work". Heating and work are ways of transferring energy; internal energy is what is stored.

Temperature and internal energy

From kinetic theory, the mean translational kinetic energy of a gas molecule is $\tfrac{3}{2}kT$: proportional to the thermodynamic temperature. The same is broadly true for solids and liquids: a higher temperature means faster random molecular motion.

So a rise in temperature always means an increase in internal energy, but an increase in internal energy does not always mean a rise in temperature.

Why this is not circular

Students sometimes find this confusing: temperature is "defined" by thermal equilibrium, yet it also measures molecular kinetic energy. Both are true. Two objects in thermal equilibrium have no net energy transfer between them; kinetic theory shows that this happens when their molecules have the same mean kinetic energy. The thermodynamic temperature scale and the molecular picture agree.

Absolute zero

At absolute zero ($0\ \text{K}$) the internal energy of a substance is at its minimum. It is not zero: the molecular potential energy is large and negative, and quantum physics shows that even the kinetic energy cannot fall entirely to zero. At A Level, it is enough to say that at absolute zero the internal energy is a minimum, and that for an ideal gas the molecules would have zero kinetic energy.

The internal energy of an ideal gas

An ideal gas has no intermolecular forces, so its molecules have no potential energy. All of its internal energy is the random kinetic energy of the molecules.

For a monatomic ideal gas (helium, neon, argon), the atoms only have translational kinetic energy, $\tfrac{3}{2}kT$ each on average. For $N$ atoms, or $n$ moles:

Molecules with more than one atom (such as $\text{O}_2$ or $\text{N}_2$) also rotate, so their internal energy is larger than $\tfrac{3}{2}nRT$, but it is still proportional to $T$ and still independent of volume. Questions that ask you to calculate $U$ will either specify a monatomic gas or give you the needed values.

The graph shows $U = \tfrac{3}{2}nRT$ for $1.0\ \text{mol}$ of a monatomic ideal gas ($U$ in J, $T$ in K): a straight line through the origin with gradient $\tfrac{3}{2}nR = 12.5\ \text{J K}^{-1}$. At $300\ \text{K}$, $U = 3.74 \times 10^{3}\ \text{J}$.

Important consequences:

• Isothermal change (constant temperature): $\Delta U = 0$ for an ideal gas, even if the pressure and volume change a lot.
• Same temperature change, same $\Delta U$: heating a gas from $300\ \text{K}$ to $400\ \text{K}$ at constant volume or at constant pressure gives the same $\Delta U$. (The energy supplied by heating is different in the two cases, because at constant pressure the gas also does work as it expands. See the first law of thermodynamics.)
• For real gases, compressing at constant temperature slightly changes the potential energy because the molecules come closer together, but for gases far from condensing this effect is very small.

Internal energy and changes of state

When a substance melts or boils, energy is supplied at constant temperature. From the table, the potential energy of the molecules rises as bonds are broken or weakened, while the mean kinetic energy stays constant.

• On melting, the molecules stay roughly as close together; only some bonds are broken. The increase in internal energy is $mL_f$ (the volume change and the work done against the atmosphere are negligible).
• On boiling, the molecules are separated completely. The internal energy increases a great deal, but not by all of $mL_v$: part of the energy supplied is used to do work pushing back the atmosphere as the vapour expands. The increase in internal energy is $mL_v - p\Delta V$.

This is a first glimpse of the first law: energy supplied by heating goes partly into internal energy and partly into work done by the system.

Worked examples

Practice questions

Answers

1. The sum of a random distribution of kinetic and potential energies of the molecules in a system.
2. Internal energy only includes the random kinetic and potential energies of the molecules. Throwing the stone gives all its molecules the same ordered motion and raises them together; the random motion and the separations of the molecules (and so the temperature) are unchanged.
3. $T = 293\ \text{K}$: $U = 1.5 \times 5.0 \times 8.31 \times 293 = 1.83 \times 10^{4}\ \text{J}$. $\Delta U = 1.5 \times 5.0 \times 8.31 \times 60 = 3.74 \times 10^{3}\ \text{J}$.
4. $U = \tfrac{3}{2}pV = 1.5 \times 2.0 \times 10^{7} \times 0.050 = 1.5 \times 10^{6}\ \text{J}$ (assuming ideal behaviour).
5. $\Delta U = mc\Delta\theta = 0.20 \times 390 \times 50 = 3.9 \times 10^{3}\ \text{J}$. The atoms vibrate with larger amplitude and higher mean speed, so their mean kinetic energy increases; their mean separation increases very slightly, so their potential energy increases slightly too.
6. (a) Temperature rises: mean kinetic energy increases; potential energy increases slightly as the liquid expands; internal energy increases. (b) Temperature constant: mean kinetic energy unchanged; molecules are separated against intermolecular forces, so potential energy increases greatly; internal energy increases.
7. No change. The internal energy of an ideal gas is all kinetic energy, which depends only on temperature; the temperature is constant. (There are no intermolecular forces, so bringing the molecules closer does not change any potential energy.)
8. Internal energy is a function of state, so the total change around the cycle is zero: $400 - 150 + \Delta U_{CA} = 0$, so $\Delta U_{CA} = -250\ \text{J}$ (a decrease of $250\ \text{J}$).
9. $W = p\Delta V = 1.01 \times 10^{5} \times 3.4 = 3.43 \times 10^{5}\ \text{J}$. Energy supplied $= mL_v = 2.0 \times 2.26 \times 10^{6} = 4.52 \times 10^{6}\ \text{J}$. $\Delta U = 4.52 \times 10^{6} - 3.43 \times 10^{5} = 4.18 \times 10^{6}\ \text{J}$. Some of the energy supplied by heating is transferred out of the system as work done pushing back the atmosphere as the steam expands.
10. (a) Total internal energy is conserved (rigid containers, so no work; insulated, so no heating from outside): $(0.40 + 0.60)T = 0.40 \times 250 + 0.60 \times 400 = 340$, so $T = 340\ \text{K}$. (b) Neon: $\Delta U = 1.5 \times 0.40 \times 8.31 \times 90 = +449\ \text{J}$. Helium: $\Delta U = 1.5 \times 0.60 \times 8.31 \times (-60) = -449\ \text{J}$. The total change is zero: energy has been transferred by heating from the helium to the neon.