Internal energy (U) – created in 1850 by Rudolf Clausius

We now come to a jump in complexity — the internal energy of a system. Internal energy is one of the most important properties in thermodynamics, and one that cannot be directly measured. It is because energy is a conserved quantity that so many thermodynamic equations work; conservation is a critical foundation of their derivation.

As covered in Chapter 3, energy occurs in two forms: kinetic and potential. Total energy is the sum of the two. Kinetic energy quantifies the motion of the parts — the atoms, the molecules, and the atoms within the molecules. Potential energy quantifies the electrical forces of attraction and repulsion between the parts, as well as between the electrons and the nucleus. These latter forces are known as chemical energy, as they are involved in chemical reactions in which electrons are rearranged. Note that the bulk kinetic energy of a moving container — a gas cylinder on a moving truck, for example — does not contribute to internal energy. Internal energy concerns only the motion and interactions of the atoms and molecules inside the system, not the motion of the system as a whole.

Clarifying energy, heat, and work – it’s all about change

Despite the fact that temperature has an absolute zero and entropy has an absolute reference point established by the Third Law (S = 0 at T = 0 for pure crystals), applied thermodynamics is fundamentally about change, not absolutes. Energy itself is not absolute. When we calculate the kinetic energy of a moving particle, we do so relative to a reference state — typically zero. It is relative speed that matters. When two atoms collide, it is their speed relative to each other that determines the outcome.

So when we talk about energy in thermodynamics, it is always the change in energy that we mean. This is why calculus plays such a dominant role in the subject — it is the natural language of change.

Heat is an especially confusing concept because we routinely use phrases such as “heat flow” and “how much heat is in that system,” which suggest that heat is a substance — a noun. As I discussed (here), there is no such thing as heat. When two systems at different temperatures make contact across a conductive boundary, their temperatures move toward each other until equilibrium is reached. The higher-temperature system experiences a decrease in energy; the lower-temperature system experiences an increase. The change in energy of each system — equal in magnitude and opposite in sign since energy is conserved — is what we call heat. More precisely: heat quantifies the change in energy of a system caused by a thermal interaction with its surroundings. Nothing flows. No substance moves. The word heat survives from a time when scientists believed otherwise, and it has confused students ever since.

Work is similar. Work is not a property stored in a system — you cannot point to it. Work quantifies the change in energy of a system caused by a mechanical process. The inward motion of a piston inside a gas-filled cylinder does work on the gas. The change in energy of the gas equals the negative change in energy of whatever drives the piston. Neither system contains work before or after — each simply has a different energy than it had before the mechanical interaction occurred.

These heat and work concepts form the basis of Rudolf Clausius’s energy balance, the first ever, in 1850 around the piston-in-cylinder assembly (here) — the device at the center of Sadi Carnot’s theoretical analysis of the heat engine. The result was the First Law of Thermodynamics:

dU = Q – W

This equation states that the change in internal energy of a system, in this case the working substance driving the piston, equals Q — the change in energy due to thermal interaction with the surroundings — minus W, the work done by the system on its surroundings. At the atomic level, both Q and W arise from the same physical event: atoms colliding at a boundary and transferring momentum. What distinguishes them is the character of the transfer — thermal on one side, mechanical on the other.

As you will see throughout the rest of this book, internal energy plays a central role throughout thermodynamics.

Reference conditions for internal energy

Tables of internal energy are always referenced to an arbitrary zero, because only differences in internal energy are physically measurable. The choice of reference depends on the substance and the application, and all such tables note the reference conditions explicitly. This is consistent with the broader theme of this book: thermodynamics works with changes, not absolutes.

Regarding the quantification of those changes, engineers often integrate heat capacity with respect to temperature to calculate ΔU. For solids and liquids, this approach works well for both ΔU and ΔH because the two are nearly equal — solids and liquids are nearly incompressible, so the PV work term that distinguishes H from U is negligible (H/enthalpy discussed in next chapter). For gases, the distinction between Cp and Cv matters and cannot be ignored, as established in Chapter 9: Cp − Cv = R per mole (ideal gas), and the volume-change work embedded in that difference must be accounted for separately depending on the conditions of the calculation.

The simplest case — and the one that connects internal energy most directly to the atomic picture — is the monatomic ideal gas. Here, there are no rotational or vibrational modes to absorb energy. Every joule added to the system goes directly into translational kinetic energy of the atoms. Internal energy is therefore directly proportional to temperature alone, and from Chapter 9’s result that the average translational kinetic energy per atom is (3/2)kT:

U = (3/2)NkT = (3/2)RT per mole, monatomic ideal gas (Nk = R)

This is the most fundamental expression for internal energy in thermodynamics — a direct, exact, atomic-level statement with no approximation involved.

The pattern extends naturally. For a diatomic ideal gas at moderate temperatures, where translational and rotational modes are active but vibration is frozen out (Chapter 9), there are five active degrees of freedom and:

U = (5/2)RT per mole, diatomic ideal gas (translation + rotation only)

For a crystalline solid, where atoms oscillate in three dimensions and both kinetic and potential energy contribute equally to each degree of freedom (Chapter 9, Dulong-Petit):

U = 3RT per mole, crystalline solid

In each case, internal energy is governed not by the mass of the atoms involved but by how many ways they can store energy — the number of active degrees of freedom. This is the same conclusion Chapter 9 reached for heat capacity, and it is no coincidence: C_v = dU/dT, so the structure of U and the structure of Cv are the same thing seen from two perspectives. Internal energy is the accumulation; heat capacity is the rate of accumulation with temperature.

END