Isothermal Process Calculator
Isothermal Process Calculator
Constant T. P₂ = P₁V₁/V₂, W = nRT ln(V₂/V₁).
Isothermal Process and Boyle's Law
An isothermal process maintains constant temperature throughout a compression or expansion. The pressure-volume relationship obeys Boyle's law (PV = constant), and the work integral evaluates to a logarithm: W = nRT ln(V₂/V₁).
Why Q = W for an Ideal Gas
An ideal gas's internal energy depends only on temperature, so an isothermal process has ΔU = 0. The first law then gives Q = W exactly: every joule of heat added becomes a joule of work done by the gas on its surroundings.
The Isothermal Curve on a PV Diagram
Plot pressure against volume for an isothermal process and the result is a smooth curve called an isotherm, following P = nRT/V. Because temperature is fixed, every point on the curve represents the same PV product. Higher-temperature isotherms sit further from the origin; each one is a hyperbola, steep at small volumes (where a small change in V produces a large change in P) and flat at large volumes.
Worked Example 1: Isothermal Expansion
1 mol of an ideal gas at 300 K expands isothermally from 1 L to 2 L. W = nRT ln(V₂/V₁) = 1 × 8.314 × 300 × ln(2) = 1729 J. The gas does 1729 J of work on its surroundings, and the same amount of heat flows in from the reservoir to keep T constant.
Worked Example 2: Isothermal Compression
The same 1 mol of gas at 300 K is compressed back from 2 L to 1 L. W = nRT ln(V₂/V₁) = 1 × 8.314 × 300 × ln(1/2) = −1729 J. The negative sign shows the surroundings do 1729 J of work on the gas, and the gas releases that same amount of heat to the reservoir to keep its temperature from rising.
See Also
Boyle's Law, Charles' Law, Isobaric Process, Isochoric Process, Combined Gas Law.