Isochoric Process Calculator

An isochoric process — also called isovolumetric or isometric — is a change in a gas's pressure and temperature that happens while its volume stays fixed. Picture a sealed, rigid steel tank: no matter how hard you heat it or cool it, the walls don't move, so the volume trapped inside never changes. Only pressure and temperature are free to respond.

Here's the key thing to notice: that direct proportion is exactly Gay-Lussac's law, P/T = constant, written for two states as P₁/T₁ = P₂/T₂. In other words, an isochoric process is the physical situation Gay-Lussac's law describes — the same equation, the same physics. If all you need is the plain P₁/T₁ = P₂/T₂ solve without any heat bookkeeping, the Gay-Lussac's law calculator handles exactly that. This page builds on top of it: once the core four values are resolved, it also works out how much heat had to be added or removed to get there, and what that heat did to the gas's internal energy.

If your numbers arrived in Celsius or Fahrenheit, convert them to kelvin first with the temperature converter — the calculator above does this automatically as you type, but it's worth knowing the absolute scale is what the underlying ratio actually depends on. And if you suspect the container isn't perfectly rigid, or that some gas might escape as it heats, the process may not be truly isochoric — the combined gas law calculator is the right tool once volume is also allowed to change.

Solving P₁/T₁ = P₂/T₂ tells you the new pressure or temperature, but it doesn't say anything about the energy involved in getting there. That's what the "Advanced: heat transfer" section above adds. Once the core four values resolve and you enter the amount of gas n in moles, the calculator applies:

The reason the calculator can stop there, without a separate work term, comes from how work is defined for an expanding or compressing gas: W = P ΔV. At constant volume, ΔV = 0, so W = 0 — no matter how large the pressure or temperature swing is, the gas does no mechanical work and none is done on it. The first law of thermodynamics, ΔU = Q − W, then collapses to something very clean:

It's worth contrasting this with the isobaric case, where pressure — not volume — is held constant. There, a heated gas is free to expand, so some of the heat you add does go into work as the gas pushes its surroundings back; only the remainder raises the internal energy. The isochoric case is the tidy opposite: with the "escape valve" of expansion removed, heat has nowhere to go but into internal energy, exactly as it does on the isothermal page's own clean fact that ΔU = 0 forces Q = W. Isochoric and isothermal end up being mirror-image "one variable does all the work" cases of the first law.

Rearranged for whichever core value is unknown: P₂ = P₁ × T₂ / T₁, P₁ = P₂ × T₁ / T₂, T₂ = T₁ × P₂ / P₁, and T₁ = T₂ × P₁ / P₂. The calculator above picks the right rearrangement automatically based on which field you leave blank, and temperatures are always converted to kelvin internally before the ratio is taken.

A sealed, rigid container being heated. Whether it's a steel cylinder, a can, or a laboratory bomb calorimeter, a fixed volume of gas trapped inside rigid walls and placed over a heat source is the textbook isochoric setup — pressure climbs in direct proportion to absolute temperature until something gives.

A pressure cooker before the valve releases. While the valve stays shut, the cooking chamber's volume is fixed, so the steam and air inside heat up isochorically, building pressure well above atmospheric until the valve finally lets some gas escape and the process stops being isochoric.

An aerosol can left in a hot car. The can's metal walls don't expand meaningfully as it warms up, so the propellant gas inside undergoes an isochoric temperature-pressure rise — which is exactly why aerosol cans carry warnings against leaving them somewhere hot.

For a broader look at the isochoric page's counterpart on the primary gas law calculator, which covers the isobaric case where pressure — not volume — is what's held fixed, see how that page's V/T solver compares to this one's P/T solver.