Isothermal Process Calculator

An isothermal process is any change in a gas's pressure or volume that happens at constant temperature — iso meaning "same" and thermal referring to heat or temperature. Nothing about the temperature itself changes; only pressure and volume are free to move, and they move in exact lockstep.

Here's the key thing to notice: that balancing relationship, holding temperature fixed while pressure and volume trade off, is Boyle's law. P₁V₁ = P₂V₂ is not a separate rule that happens to apply to isothermal processes — it is the isothermal gas law, derived directly from the requirement that temperature not change. Boyle's law is exactly this isothermal relationship — for the plain P₁V₁ = P₂V₂ solve without the energy bookkeeping, use the Boyle's law calculator. What this page adds on top is the other half of the picture: how much energy moves as work while pressure and volume swap places, which the plain Boyle's law solve doesn't report.

It's worth contrasting this with the constant-pressure case. If pressure were held fixed instead of temperature, you'd be looking at an isobaric process instead — that's exactly what the calculator for Charles' law handles. Isothermal and isobaric are mirror images of each other: one variable is pinned down, and the calculator solves for how the other two trade off.

Solving P₁V₁ = P₂V₂ tells you the new pressure or volume, but it doesn't say anything about the energy involved in getting there. That's what the "Advanced: work done" section of the calculator above is for. Once you supply the constant temperature T and the amount of gas n, the work done by the gas as it moves from V₁ to V₂ is:

The sign of the result carries real meaning, not just bookkeeping. When the gas expands (V₂ > V₁), the ratio V₂/V₁ is greater than 1, its natural log is positive, and W comes out positive — the gas is doing work on whatever is pushing back against it, like a piston. When the gas is compressed (V₂ < V₁), the ratio is less than 1, the log is negative, and W comes out negative — work is being done on the gas by an outside force, rather than by the gas itself.

There's a neat consequence hiding in the fact that temperature never changes. For an ideal gas, internal energy is a function of temperature alone — nothing else about the gas's state affects it. Since ΔT = 0 for an isothermal process, ΔU = 0 as well. Plug that into the first law of thermodynamics, ΔU = Q − W, and it collapses to Q = W: whatever work the gas does while expanding must be supplied entirely as heat flowing in, and whatever work is done compressing the gas must be released entirely as heat flowing out. None of the energy goes toward warming or cooling the gas itself, because by definition, it can't.

That's a genuinely different situation from a process where temperature is allowed to drift. If your scenario changes pressure, volume, and temperature together, this page's assumptions no longer apply — reach for the combined gas law calculator instead, which handles all three variables changing at once rather than assuming one of them is pinned.

No real process is perfectly isothermal, but plenty come close enough for the maths above to be a good approximation. The common ingredient is time: a gas compressed or expanded slowly, while sitting in good thermal contact with something much larger than itself — a heat reservoir, in the textbook phrase. If a piston is pushed in gradually enough, any heat generated by the compression has time to leak out into the surrounding air or a water bath before the temperature can climb, and the process stays close to isothermal from start to finish. Slow gas compression or expansion inside a well-insulated laboratory apparatus, submerged in a temperature-controlled bath, is about as close to textbook-isothermal as real equipment gets.

Contrast that with a rapid change — a piston slammed in over a fraction of a second, or a gas escaping through a valve in a sudden burst. There, the change happens far faster than heat can flow in or out, so the process is closer to the opposite extreme: an adiabatic process, where no heat exchange happens at all and temperature swings sharply instead. That's a different calculator problem entirely — worth knowing about as a contrast, but not something this page's constant-temperature maths applies to.

If your two known volumes happen to be recorded in mismatched units, run them through the volume converter first, so you can sanity-check the ratio in the work formula by eye before trusting the answer above.