Internal Energy, Temperature and Heat

Every chapter of this course so far has treated a moving object as a single point with one velocity and one kinetic energy. Thermal physics abandons that simplification. A litre of water is $3.3\times 10^{25}$ molecules, each with its own velocity, colliding billions of times a second; "the temperature of the water" and "the heat supplied to it" are statements about that swarm, not about any one molecule. This chapter builds the three quantities that the rest of the block depends on — internal energy, temperature and heat — and, just as importantly, keeps them apart, because conflating them is the single largest source of error in the whole topic.

The strategy is to define each quantity twice: once macroscopically, in terms of what a thermometer or a joule-meter reads, and once microscopically, in terms of the molecules. The two pictures must agree, and forcing them to agree is what fixes the value of Boltzmann's constant and gives the phrase "absolute zero" a precise meaning.

Reading note. Work through the two worked examples in Section 7 with a calculator. The numbers — a molecular speed of a kilometre per second, an internal energy of a few kilojoules in half a mole of gas — are worth carrying in your head as sanity checks for the rest of the block.

1. Thermal Equilibrium and the Zeroth Law

Put a warm block of metal in contact with a cold one. Energy flows from the warm block to the cold one, never the reverse, until some property common to both stops changing. At that point the two blocks are in thermal equilibrium, and the shared value of that property is what we call temperature.

Definition. Two bodies are in thermal equilibrium when there is no net transfer of energy between them when they are placed in thermal contact. Temperature is the quantity that is equal for two bodies in thermal equilibrium, and that decides the direction of net energy flow when they are not: energy always flows, on balance, from higher temperature to lower.

This is not quite as trivial as it sounds, because it contains a hidden assumption that has to be promoted to a law:

The Zeroth Law of Thermodynamics. If body $A$ is in thermal equilibrium with body $C$, and body $B$ is also in thermal equilibrium with $C$, then $A$ and $B$ are in thermal equilibrium with each other.

Without the zeroth law, thermometry would be impossible. A thermometer is just a body $C$ with a conveniently readable property (the length of a mercury column, the resistance of a platinum wire, the pressure of a fixed volume of gas). We bring it to equilibrium with $A$, read it, bring it to equilibrium with $B$, read it — and the zeroth law is the guarantee that equal readings mean $A$ and $B$ would themselves be in equilibrium if placed in contact.

2. Temperature Scales and Absolute Zero

The Celsius scale fixes its zero at the freezing point of water and $100$ at its boiling point (at standard pressure). It is a scale of convenience, and its zero has no special physical status.

The thermodynamic or Kelvin scale, which is the SI scale and the only one valid in the equations of this block, fixes its zero at absolute zero: the temperature at which the random molecular kinetic energy reaches its minimum possible value and can be reduced no further. On this scale the two scales run in step — one kelvin is exactly one degree Celsius in size — so conversion is a shift, not a stretch:

For most calculations $273$ is close enough. Two consequences matter constantly:

• A temperature difference has the same numerical value in kelvin and in degrees Celsius: $\Delta T = 20\ \text{K} = 20\ ^{\circ}\text{C}$. So $\Delta T$ may be computed in either scale.
• An absolute temperature must be in kelvin in every equation that follows — the gas laws, the kinetic-theory result $\bar{E}_k = \tfrac{3}{2}k_BT$, everything. Substituting $25\ ^{\circ}\text{C}$ where $298\ \text{K}$ is required is the most common single mistake in thermal physics.

Absolute zero is $0\ \text{K} = -273.15\ ^{\circ}\text{C}$. It is unattainable in practice, but the linear fall of gas pressure with temperature at fixed volume extrapolates cleanly to zero pressure there, which is the experimental route to locating it.

3. The Microscopic Picture: What Internal Energy Is

The molecules of any substance are in ceaseless random motion and exert electrical forces on one another. Two kinds of energy follow:

• the kinetic energy of the random translational, rotational and vibrational motion of the molecules;
• the potential energy stored in the intermolecular forces — the "springs" between molecules that resist being stretched or compressed.

Definition. The internal energy $U$ of a body is the total of the random kinetic energies of its molecules and the potential energies of the interactions between them. It does not include the ordered kinetic energy of the body moving as a whole, nor its gravitational potential energy in an external field.

A cricket ball flying through the air has kinetic energy $\tfrac{1}{2}mv^2$ as a whole and, quite separately, an internal energy of order $10^{5}\ \text{J}$ locked in the thermal motion of its molecules. Bringing the ball to rest against a wall converts some of the first into the second: the ball warms up.

The relative size of the two contributions depends on the phase:

• In a solid, molecules are locked into a lattice. The kinetic part is vibrational; the potential part is large and negative, because the molecules sit deep in their mutual potential wells.
• In a liquid, molecules are still close-packed but mobile; the potential part is somewhat less negative.
• In an ideal gas, the molecules are so far apart that intermolecular forces — and hence the intermolecular potential energy — are negligible. The internal energy of an ideal gas is therefore purely molecular kinetic energy, and this is what makes the ideal gas the clean case treated in Chapters 4 and 5.

4. Temperature as a Measure of Mean Molecular Kinetic Energy

The microscopic meaning of temperature is the bridge between the two pictures. For an ideal gas, the kinetic theory of Chapter 4 will show — and it is quoted here because the rest of this chapter needs it — that the mean random translational kinetic energy of one molecule depends only on the absolute temperature:

where $k_B = 1.38\times 10^{-23}\ \dfrac{\text{J}}{\text{K}}$ is Boltzmann's constant. Three features of this equation are worth pausing on.

• It contains no reference to the mass of the molecule or the amount of substance. At a given temperature a helium atom and a xenon atom have the same mean kinetic energy; the heavier one simply moves more slowly. This is why temperature, unlike internal energy, is an intensive quantity — doubling the amount of gas does not double its temperature.
• It is a statement about a mean. Individual molecules have a wide spread of speeds (the Maxwell–Boltzmann distribution); $\bar{E}_k$ is their average.
• Because $k_B > 0$, the mean kinetic energy falls as $T$ falls and would reach zero at $T = 0\ \text{K}$. That is the precise content of "absolute zero is the temperature of minimum molecular energy."

For a monatomic ideal gas of $N$ molecules — the case the IB syllabus uses, where all the molecular kinetic energy is translational — the internal energy is just $N$ times this mean:

using $N = nN_A$ and the molar gas constant $R = N_A k_B = 8.31\ \dfrac{\text{J}}{\text{mol}\cdot\text{K}}$. The internal energy of an ideal gas depends on temperature and nothing else — not on its volume or pressure separately. (For a diatomic gas the molecules also store rotational kinetic energy, and $U$ is larger — $\tfrac{5}{2}nRT$ near room temperature — but still depends on temperature alone.)

5. Heat and Work: The Two Ways to Change Internal Energy

The internal energy of a body can be changed in exactly two ways.

Definition. Heat $Q$ is energy transferred between a body and its surroundings because of a temperature difference between them. Work $W$ is energy transferred by any other means — typically a force moving through a distance, such as a gas being compressed or a liquid being stirred.

Both are measured in joules, and both describe energy in transit. A body does not "contain heat" any more than a bank account contains "deposits": once the transfer is complete, the energy that arrived as heat is indistinguishable from the energy that arrived as work — it is simply part of the internal energy. Phrases like "the heat in the gas" are therefore meaningless, and examiners penalise them.

Conservation of energy then gives the first law of thermodynamics:

with the sign convention that $Q$ is positive when heat is added to the body and $W$ is positive when work is done on the body. (Some texts write $Q = \Delta U + W_{\text{by}}$, where $W_{\text{by}} = -W$ is the work done by the body; the physics is identical, and only the bookkeeping differs.) Joule's paddle-wheel experiment is the classic demonstration that the two routes are equivalent: churning water with a falling weight raises its temperature by exactly the amount that supplying the same number of joules as heat would.

6. Telling Temperature, Internal Energy and Heat Apart

The three quantities are routinely confused because everyday language runs them together. A clean set of distinctions:

• Temperature $T$ (unit $\text{K}$) measures the mean random kinetic energy per molecule. It is intensive: it does not depend on how much substance is present, and it decides the direction of net energy flow.
• Internal energy $U$ (unit $\text{J}$) is the total random kinetic energy plus the total intermolecular potential energy. It is extensive: double the amount of substance at the same temperature and $U$ doubles.
• Heat $Q$ (unit $\text{J}$) is not a property of a body at all. It is the energy transferred between a body and its surroundings because of a temperature difference — a process, complete once the transfer ends.

Two standard thought-experiments make the distinctions bite. A spark from a grinder is at over $1000\ \text{K}$ but carries so little internal energy — it is a speck of metal — that it does not hurt when it lands on skin. A bath of water at only $320\ \text{K}$ — hundreds of kelvin cooler than the spark — contains vastly more internal energy and would scald. And two blocks, one of copper and one of water, of equal mass at the same temperature, have very different internal energies, because storing energy in water raises its temperature far less per joule than storing it in copper does (Chapter 2).

7. Worked Examples

Example 1 — the speed of a gas molecule. Estimate the mean translational kinetic energy of a helium atom at $20\ ^{\circ}\text{C}$, and the corresponding root-mean-square speed. Take the mass of a helium atom as $4.00\,u$ with $u = 1.66\times 10^{-27}\ \text{kg}$.

First convert the temperature: $T = 20 + 273 = 293\ \text{K}$. Then

The atomic mass is $m = 4.00(1.66\times 10^{-27}) = 6.64\times 10^{-27}\ \text{kg}$. Setting $\tfrac{1}{2}m\overline{c^{2}} = \bar{E}_k$,

Room-temperature gas molecules move at roughly the speed of a rifle bullet. A heavier molecule at the same temperature — nitrogen, say, seven times the mass — moves $\sqrt{7}\approx 2.6$ times slower, near $510\ \dfrac{\text{m}}{\text{s}}$.

Example 2 — internal energy of a gas sample. A flask holds $0.50\ \text{mol}$ of argon, which behaves as a monatomic ideal gas, at $27\ ^{\circ}\text{C}$. Find its internal energy, and the energy that must be supplied to raise its temperature to $127\ ^{\circ}\text{C}$ at constant volume.

With $T_1 = 300\ \text{K}$,

At $T_2 = 400\ \text{K}$, $U_2 = \tfrac{3}{2}(0.50)(8.31)(400) = 2.49\times 10^{3}\ \text{J}$, so

At constant volume no work is done on or by the gas, so by the first law $Q = \Delta U = 6.2\times 10^{2}\ \text{J}$, supplied entirely as heat and appearing entirely as extra molecular kinetic energy. For a real solid warmed through the same $100\ \text{K}$, part of the supplied energy would instead go into intermolecular potential energy, and $\Delta U$ would not be simply proportional to $T$.

Common Exam Pitfalls

• Using degrees Celsius where kelvin is required. Any equation with $T$ standing alone — $\bar{E}_k = \tfrac{3}{2}k_BT$, $U = \tfrac{3}{2}nRT$, and every gas law to come — needs absolute temperature. Only differences $\Delta T$ may be left in Celsius.
• Treating heat as a property of a body. Heat is energy in transit due to a temperature difference; once transferred it is just internal energy. "The heat contained in the gas" is a marked-wrong phrase.
• Putting the whole body's motion or its height into $U$. Internal energy is the energy of the random molecular motion and the intermolecular forces only — not $\tfrac{1}{2}mv^2$ of the body and not $mgh$.
• Assuming equal temperature implies equal internal energy. Two bodies of different mass, or different material, at the same temperature generally hold very different internal energies.
• Assuming internal energy is proportional to temperature for any substance. This holds for an ideal gas, where $U = \tfrac{3}{2}nRT$. For solids and liquids the intermolecular potential energy also changes with temperature.
• Saying "cold flows into" a warm body. There is no such thing as a flow of cold; there is only a net transfer of energy out of the warmer body.

Summary

• Two bodies are in thermal equilibrium when no net energy passes between them; temperature is what they then share, and it sets the direction of net energy flow otherwise. The zeroth law makes thermometry possible.
• The Kelvin scale is the SI scale: $T/\text{K} = \theta/^{\circ}\text{C} + 273.15$. A difference $\Delta T$ is the same number in K and $^{\circ}$C; an absolute $T$ must be in K in every equation of this block.
• Absolute zero, $0\ \text{K} = -273.15\ ^{\circ}\text{C}$, is the temperature of minimum random molecular energy.
• Internal energy $U$ = total random molecular kinetic energy + total intermolecular potential energy. It excludes the ordered motion of the body and its external potential energy.
• For an ideal gas the intermolecular potential energy is negligible, so $U$ is a function of temperature alone; for a monatomic ideal gas $U = \tfrac{3}{2}Nk_BT = \tfrac{3}{2}nRT$.
• The mean translational kinetic energy per molecule is $\bar{E}_k = \tfrac{3}{2}k_BT$, with $k_B = 1.38\times 10^{-23}\ \dfrac{\text{J}}{\text{K}}$ and $R = N_Ak_B$.
• Heat $Q$ and work $W$ are the two ways to change internal energy, related by the first law $\Delta U = Q + W$ ($Q$ into the body positive, $W$ on the body positive).