Internal Energy
Before a single equation of thermodynamics can be written down, you need a precise answer to a deceptively simple question: where exactly is the energy inside a gas actually stored? Getting this straight is what makes the whole of thermodynamics click into place.
What internal energy actually is
Every thermodynamic system — a cylinder of trapped gas in an engine, the air in a bicycle pump, the working fluid of a refrigerator — is really just an enormous population of individual particles, each one jostling around with its own share of energy. The internal energy $U$ of a system is defined as the sum of the random kinetic and potential energies of all the particles that make it up.
Two words in that definition are doing all the work: random and sum. Internal energy only counts the microscopic, disordered motion and interaction energy of individual particles relative to each other — it has nothing to do with any large-scale, organised motion or position of the system as a whole. A canister of gas sitting on a shelf and the same canister flying through the air at 300 m/s have exactly the same internal energy, even though the flying canister obviously carries a great deal more kinetic energy in total. That extra energy is bulk kinetic energy — ordered motion of the whole system together — and it is kept entirely separate from $U$. Likewise, lifting a gas cylinder to the top of a building increases its gravitational potential energy as a whole object, but does nothing to the internal energy of the gas inside it. Internal energy is about the chaotic, particle-by-particle world inside the system, not about where the system as a whole happens to be or how fast it's travelling.
- Random kinetic energy — the translational (and, for molecules, rotational and vibrational) kinetic energy of individual particles due to their own thermal motion, in every direction, cancelling out on average so there's no overall drift.
- Random potential energy — energy stored in the interactions and arrangement between particles, for example the intermolecular bonds that must be stretched or broken as a liquid evaporates into a gas.
Why an ideal gas makes this simple
In the kinetic theory model of an ideal gas, particles are treated as point-like and as exerting no forces on one another except during the instant of collision. With no long-range intermolecular forces to speak of, there is essentially no potential energy stored between the particles of an ideal gas — the entire internal energy is just the sum of the particles' random kinetic energies.
This matters enormously, because the average random kinetic energy of the particles in a gas is set entirely by its absolute temperature: raise the temperature and the particles move faster on average, storing more kinetic energy; lower it and they slow down. Since an ideal gas has no potential energy contribution at all, this means:
For an ideal gas, internal energy depends only on temperature. There is no other variable — not pressure, not volume individually — that changes $U$ on its own. This means a change in an ideal gas's temperature is the direct, reliable signature of a change in its internal energy, and conversely, if the temperature doesn't change, neither does the internal energy, no matter what else is happening to the gas's pressure or volume. Every calculation involving $\Delta U$ in the rest of this topic — the first law, isothermal processes, adiabatic processes — ultimately traces back to this one fact.
This is precisely the link that makes it possible to talk about "gaining" or "losing" internal energy in the following lessons purely in terms of temperature change, without having to track the motion of individual molecules directly.
The equation of state: connecting the macroscopic variables
While internal energy itself is a statement about microscopic particle energies, everything you actually measure and control in a real engine or gas system — pressure $p$, volume $V$, and temperature $T$ — is macroscopic. The relationship linking these bulk quantities together for a fixed amount of ideal gas is the equation of state:
where $n$ is the amount of gas present in moles, and $R$ is the molar gas constant ($8.31\ \text{J K}^{-1}\text{mol}^{-1}$). Temperature $T$ must always be in kelvin for this equation to hold, since it derives from the absolute (kinetic-energy) scale of temperature, not the Celsius scale.
This single equation is the tool that will be used again and again throughout this topic to track how a gas's pressure, volume and temperature change together as it's compressed, expanded, heated or cooled — whether that gas is the working fluid inside a petrol engine cylinder, a diesel engine, a refrigerator, or any other thermodynamic system built around a trapped gas.
Setting up the first law
With internal energy properly defined, and with $pV = nRT$ in hand as the equation connecting the macroscopic variables, the stage is set for the single most important relationship in this whole topic: the first law of thermodynamics, which describes exactly how a gas's internal energy changes when heat is transferred to or from it and when it does work by expanding or being compressed. That's the subject of the next lesson — and everything covered here, particularly the fact that $\Delta U$ for an ideal gas is a story purely about temperature change, is what makes that law usable in practice.
Key takeaways
- Internal energy $U$ is the sum of the random kinetic and random potential energies of the particles making up a system.
- Internal energy excludes any bulk (ordered) kinetic energy of the system moving as a whole, and any bulk gravitational potential energy of its position — only the random, microscopic motion and interactions count.
- In the ideal gas model, particles have no significant intermolecular potential energy, so internal energy is entirely random kinetic energy.
- For an ideal gas, internal energy depends only on temperature — no other variable changes $U$ independently.
- A temperature change is therefore the direct, reliable signature of a change in an ideal gas's internal energy.
- The equation of state $pV = nRT$ links the macroscopic variables pressure, volume, temperature and amount of gas, and underpins the calculations throughout this topic.
- These ideas set up the first law of thermodynamics, covered next, which describes how $U$ changes under heating and work.