Isochoric Process
An isochoric (isovolumetric) process is a change of state at constant volume. Because no expansion work is done (W = ∫p dV = 0), the first law gives Q = ΔU, so heat at constant volume measures the change in internal energy and defines Cv. For an ideal gas, pressure is proportional to absolute temperature, p/T = constant (the pressure law associated with Amontons and Gay-Lussac).
Curious
A sealed spray can, a pressure cooker before its valve opens, and a car tyre on a hot road are all roughly constant-volume containers. Warm them and the pressure rises in proportion to the absolute temperature (see Ideal Gas Law). That is why aerosol cans carry warnings not to expose them to heat.
Because the gas does no work, the heat supplied all becomes internal energy. Chemists exploit this in the bomb calorimeter, a sealed steel vessel in which burning a sample at constant volume measures the internal-energy change of the reaction.
Exploring
| Process | Held constant | Ideal-gas results |
|---|---|---|
| Isothermal | T | Q = W = nRT ln(V2/V1), ΔU = 0 |
| Adiabatic | Q = 0 | pVγ = const, W = −ΔU |
| Isochoric | V | W = 0, Q = ΔU = nCvΔT |
| Isobaric | p | W = pΔV, Q = nCpΔT |
Deep Dive
Thermodynamics. At constant volume, dU = δQ = CvdT, so Cv = (∂U/∂T)V. The entropy change is ΔS = nCv ln(T2/T1). On a p–V diagram an isochore is a vertical line. Isochoric heating and cooling steps appear in the ideal Otto cycle (see Heat Engines) and the Stirling cycle.
Calorimetry. A bomb calorimeter measures ΔU of combustion; converting to the enthalpy change uses ΔH = ΔU + ΔngasRT (see Enthalpy). Food energy values were historically determined this way, with corrections for digestibility.
History. The proportionality of gas pressure to temperature at fixed volume was observed by Guillaume Amontons around 1700 and later quantified with greater care; attributions to Amontons, Charles and Gay-Lussac vary between textbooks, and the names “Gay-Lussac's law” and “Amontons's law” are both used for it. Extrapolating the linear relation to zero pressure gave an early estimate of absolute zero near −273 °C (see Temperature).
Concept Relationship Map
Requires: first law, ideal gas law
Enables: Cv, bomb calorimetry, Otto cycle
Commonly confused with: isobaric process
Sources
- Gay-Lussac, J.L. (1802). Annales de Chimie, 43, 137–175.
- Çengel, Y.A., Boles, M.A. (2015). Thermodynamics: An Engineering Approach (8th ed.). McGraw-Hill.
- Schroeder, D.V. (2000). An Introduction to Thermal Physics. Addison-Wesley.