How this instrument works
Internal energy, here, is the total kinetic energy locked up in the random motion of every molecule in a sample of gas — not the energy of the gas moving as a whole, like a jet of steam from a nozzle, but the microscopic jostling that persists even when the container sits perfectly still. For a monatomic ideal gas — helium, neon, argon, any gas whose particles behave as featureless points with no internal structure to spin or flex — the equipartition theorem assigns exactly (1⁄2)RT of energy to each of the three directions a particle can move: up-down, left-right, forward-back. Three directions times (1⁄2)RT per mole gives the (3⁄2)RT that multiplies against the amount of gas, n, to produce U = (3⁄2)nRT.
The formula depends on temperature alone — not on pressure, not on volume, not on how tightly the gas is squeezed into its container. That is not an approximation; it follows from the ideal-gas model itself, where molecules are assumed not to attract or repel one another, so pushing them closer together stores no extra potential energy between them. James Joule tested exactly this in 1845 by letting a gas expand freely into a vacuum: the temperature barely moved, evidence that internal energy rides on temperature and nothing else, at least for a gas idealized enough to ignore intermolecular forces.
The 3⁄2 factor is specific to monatomic gases, which is why this instrument should not be pointed at air, nitrogen, or carbon dioxide. A diatomic molecule like N₂ can also rotate about two axes, adding a further RT and making its internal energy (5⁄2)nRT; add vibration at high temperature and the multiplier climbs again. Real monatomic gases at ordinary pressures track the ideal formula closely — a genuinely good approximation for helium in a balloon or argon in a welding cylinder — but it drifts at the extreme pressures and cryogenic temperatures where intermolecular forces finally start to matter.
- Enter the quantity of gas into Amount of gas, mol — moles, not grams; look up the molar mass to convert if your data is in mass.
- Set Gas temperature in °C, or switch the unit menu to kelvin if that is how your data arrives — the instrument adds 273.15 internally either way.
- Read Internal (thermal) energy in joules, or flip the unit menu to kilojoules for larger samples.
- Remember the 3⁄2 multiplier assumes a monatomic gas — helium, neon, argon, krypton, xenon — not air or any diatomic mixture.
Worked example — 1 mole of helium at 25 °C
Take 1 mole of helium — Amount of gas, mol = 1 — sitting at room temperature, Gas temperature = 25 °C. Converting to kelvin first: 25 + 273.15 = 298.15 K. The formula then runs straight through: U = 1.5 × 1 × 8.314462618 × 298.15 = 3718.43554434 J, which the Internal (thermal) energy field rounds for display but keeps in full precision underneath — about 3,718 J, or 3.718 kJ.
That figure is entirely separate from any heat needed to warm the helium up to 25 °C from some colder start — this instrument reports the energy the gas holds at its current state, not the energy spent getting there, which is why the same 3,718 J shows up whether the helium arrived at room temperature by heating from −20 °C or by cooling from 200 °C. Doubling the amount to 2 moles at the same temperature exactly doubles the result to 7,436.87 J, since n sits alone and unsquared in the formula.
Questions
What is the difference between internal energy and heat?
Internal energy, U, is a property of the gas's current state — how much it holds right now — while heat, Q, is energy in transit between two states, like the amount added while warming from one temperature to another. A sample at 25 °C has one value of U regardless of its history, but the heat needed to reach 25 °C depends on the path: how cold it started, whether it did work along the way, and what process carried it there.
Why is the multiplier 3⁄2 instead of some other fraction?
It comes from the equipartition theorem: a monatomic gas particle can only move in three independent directions — no internal spin or vibration to store energy in — and each direction gets an equal share of (1⁄2)RT per mole. Three directions times that share is (3⁄2)RT, the multiplier this instrument uses. A diatomic gas that can also rotate adds more terms and a larger fraction.
Does this formula work for air, nitrogen, or carbon dioxide?
No, not accurately. Those are diatomic or polyatomic molecules that can rotate, and at higher temperatures vibrate, storing extra energy the 3⁄2 factor does not count. Nitrogen and oxygen, the bulk of air, run closer to (5⁄2)nRT near room temperature because of two extra rotational degrees of freedom. This instrument is built for monatomic gases only — helium, neon, argon, krypton, xenon, radon.
Why doesn't pressure or volume appear in the formula?
Because an ideal gas's molecules are assumed to exert no forces on one another, so there is no potential energy stored between them that squeezing or expanding the gas could change — only temperature, which tracks their kinetic energy, matters. James Joule demonstrated this directly in 1845 by letting gas expand into a vacuum and finding almost no temperature change, evidence that internal energy depends on T alone for a gas idealized this way.
What happens to internal energy at absolute zero?
It falls to exactly 0 J. Set Gas temperature to −273.15 °C — absolute zero on the Celsius scale — and T in kelvin becomes 0, so U = (3⁄2)nR × 0 vanishes regardless of how much gas is present. Classically this means molecular motion stops entirely; real gases never actually reach this point, and quantum mechanics revises the picture slightly at the very bottom, but the ideal formula treats it as a clean floor.
How does this relate to the average energy of a single molecule?
Divide U by the number of molecules and the same 3⁄2 factor reappears at the microscopic scale: each molecule averages (3⁄2)kT of kinetic energy, where k is the Boltzmann constant, 1.380649×10⁻²³ J/K. Multiply that per-molecule figure by Avogadro's number and the mole-scale and molecule-scale versions of the formula agree exactly.