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 report values measured from a chosen reference state, placed wherever is convenient for the substance and application. The numbers themselves therefore carry no physical meaning on their own; only differences between them do. This is the broader theme of this book: thermodynamics works with changes.

To calculate those changes, engineers integrate heat capacity over temperature. For internal energy, the relevant heat capacity is Cv:

ΔU = ∫C_v dT

For an ideal gas, this holds for any process, not just one at constant volume, because an ideal gas’s internal energy depends on temperature alone. For enthalpy, the relevant heat capacity is C_p. For solids and liquids the distinction hardly matters: they are nearly incompressible, so the PV term that separates H from U is negligible and ΔU ≈ ΔH (enthalpy is taken up in the next chapter). For gases it matters a great deal. As established in Chapter 9, C_p − C_v = R per mole for an ideal gas, and that difference is the work the gas does in expanding at constant pressure. In every case, these integrals apply within a single phase; a phase change adds its own energy, which must be included separately.

The simplest case, and the one that connects internal energy most directly to the atomic picture, is the monatomic ideal gas. With no rotation or vibration available, all of its internal energy is translational kinetic energy. From Chapter 9, the average translational kinetic energy per atom is (3/2)k_BT, so:

U = (3/2)Nk_BT = (3/2)nRT

measured from atoms at rest relative to the system’s center of mass and not interacting. Within that reference, the result is exact, a direct atomic-level statement with no approximation.

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

U = (5/2)nRT

For a crystalline solid, the atoms oscillate in three dimensions, with kinetic and potential energy each contributing to every direction of vibration (Chapter 9, Dulong–Petit):

U = 3nRT + U₀

Here U₀ is the potential energy that binds the crystal, measured from infinite separation. It is negative, because the atoms sit lower in energy together than apart, and it is present even at absolute zero. The 3nRT term holds only at high temperatures; at lower temperatures, quantum effects reduce it, as Chapter 9 describes.

In each case, the temperature-dependent part of the internal energy is governed not by the mass of the atoms 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. Since C_v = (∂U/∂T)_V, internal energy and heat capacity are the same structure seen from two perspectives: internal energy is the accumulation, and heat capacity is the rate of accumulation with temperature.

END