Thermodynamic Processes Calculator — Ideal Gas P-V Diagram
Analyse any of the four standard ideal-gas processes — isothermal, isobaric, isochoric, or adiabatic — by entering the initial state and one constraint. Get the final state variables (P₂, V₂, T₂), work done, heat transferred, and internal energy change, with a live P-V diagram.
Process type
kPa
L
K
L
Process: Isothermal (T = const)
- 1
P₁V₁ = nRT₁
101.325 × 10³ × 1 × 10⁻³ = 101.325 J - 2
ln(V₂ ÷ V₁)
ln(2 ÷ 1) = 0.69315 - 3
Work W = P₁V₁ × ln(V₂/V₁)
101.325 × 0.69315 = 70.23
How does this calculator work?
For an ideal gas, four classic processes: isothermal (T const, W = nRT·ln(V₂/V₁)), isobaric (P const, W = PΔV), isochoric (V const, W = 0), and adiabatic (Q = 0, PVᵞ = const). Enter initial state P₁, V₁, T₁ and the final volume or pressure; get P₂, T₂, work W, heat Q, and ΔU with a P-V diagram.
Formula
How this is calculated
An ideal gas obeys PV = nRT. A thermodynamic process constrains one or two variables while the rest change.
**Isothermal (T = const):** Temperature is fixed (by a heat reservoir), so PV = nRT₁ = const, giving P₂ = P₁V₁/V₂. The work done by the gas is W = nRT·ln(V₂/V₁) = P₁V₁·ln(V₂/V₁). Internal energy is unchanged (ΔU = 0 for an ideal gas at constant T), so all heat Q = W must come from the reservoir.
**Isobaric (P = const):** Pressure is fixed. From Charles's law V/T = const, so T₂ = T₁·V₂/V₁. Work W = P·ΔV = P(V₂ − V₁). Heat and internal energy changes involve the molar heat capacities Cv = R/(γ−1) and Cp = γR/(γ−1): ΔU = nCv·ΔT, Q = nCp·ΔT.
**Isochoric (V = const):** No work is done (W = 0). From Gay-Lussac's law P/T = const, so T₂ = T₁·P₂/P₁. All heat goes into internal energy: Q = ΔU = nCv·ΔT.
**Adiabatic (Q = 0):** No heat exchange. The constraint PVᵞ = const gives P₂ = P₁(V₁/V₂)ᵞ and T₂ = T₁(V₁/V₂)ᵞ⁻¹. Work W = (P₁V₁ − P₂V₂)/(γ−1) and ΔU = −W.
Frequently asked questions
γ = Cp/Cv is the ratio of molar heat capacities. For a monoatomic ideal gas (He, Ar) γ = 5/3 ≈ 1.667. For diatomic gases (N₂, O₂, air) at room temperature γ = 7/5 = 1.4. For triatomic or polyatomic gases (CO₂, H₂O vapour) γ ≈ 1.33. Use 1.4 for most air problems.
On a P-V plot, an isothermal is P ∝ 1/V (hyperbola with slope −P/V). An adiabatic is P ∝ V^(−γ) (slope −γP/V). Since γ > 1, the adiabatic slope is steeper — pressure drops more quickly with volume than during an isothermal expansion, because the gas also cools as it expands with no heat input.
No. The formulas assume an ideal gas (no intermolecular forces, point-like molecules). Real gases deviate at high pressures or near phase boundaries. The van der Waals equation or other equations of state are needed for accurate real-gas work — this calculator is for ideal-gas approximations used in physics and engineering education.
Also known as
TG we-Calculate Editorial Team. (2026). Thermodynamic Processes Calculator — Ideal Gas P-V Diagram [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/thermodynamic-processes-calculator
TG we-Calculate Editorial Team. "Thermodynamic Processes Calculator — Ideal Gas P-V Diagram." TG we-Calculate. 2026. https://we-calculate.com/calculator/thermodynamic-processes-calculator.
TG we-Calculate Editorial Team, "Thermodynamic Processes Calculator — Ideal Gas P-V Diagram," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/thermodynamic-processes-calculator
@misc{wecalculate_thermodynamic_processes_calculator, title = {Thermodynamic Processes Calculator — Ideal Gas P-V Diagram}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/thermodynamic-processes-calculator}}, year = {2026}, note = {TG we-Calculate} }
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