ThermodynamicsW = ∫P dV

PV Diagram Explorer

Plot thermodynamic processes on a PV diagram. Compare isothermal, adiabatic, isobaric, and isochoric processes and calculate work done.

— Isothermal (blue)— Adiabatic (red)— Isobaric (orange)— Isochoric (green)

Parameters

Initial Pressure P₁

atm

Initial Volume V₁

Final Volume V₂

Process Type

P₁2.00 atm

V₁5.00 L

T₁121.9 K

P₂1.00 atm

V₂10.00 L

T₂121.9 K

W702.3 J

ΔU0.0 J

Q702.3 J

About PV Diagrams

A PV diagram (pressure-volume diagram) plots the state of a gas as it undergoes a thermodynamic process. The horizontal axis is volume V and the vertical axis is pressure P. The area under any curve on a PV diagram equals the work done by the gas during that process.

ℹFirst Law of Thermodynamics: ΔU = Q − W. The change in internal energy equals heat added to the gas minus work done by the gas.

The Four Basic Processes

Process	Constant Quantity	PV Curve Shape	Work W	Heat Q	ΔU
Isothermal	Temperature T	Hyperbola (PV = const)	nRT ln(V₂/V₁)	= W	0
Adiabatic	Heat Q = 0	Steeper hyperbola (PV^γ = const)	(P₁V₁−P₂V₂)/(γ−1)	0	= −W
Isobaric	Pressure P	Horizontal line	PΔV	nCpΔT	nCvΔT
Isochoric	Volume V	Vertical line	0	nCvΔT	= Q

Isothermal Process

At constant temperature, the ideal gas law gives PV = nRT = constant. The gas expands along a hyperbola. Work done by the gas equals the area under the curve:

W_{iso} = n R T ln (\frac{V _{2}}{V _{1}})

Adiabatic Process

No heat is exchanged with the surroundings. The relation PVᵞ = constant applies, where γ = Cp/Cv is the heat capacity ratio (5/3 for monatomic ideal gas, 7/5 for diatomic).

P_{2} = P_{1} (\frac{V _{1}}{V _{2}})^{γ}, W = \frac{P _{1} V _{1} - P _{2} V _{2}}{γ - 1}

💡The adiabatic curve is always steeper than the isothermal curve through the same point, because adiabatic expansion also lowers temperature, causing a greater pressure drop.

Worked Example — Isothermal Expansion

1 mol of gas at T = 300 K expands from V₁ = 5 L to V₂ = 10 L at constant temperature:

W = n R T ln (\frac{V _{2}}{V _{1}}) = 1 \times 8.314 \times 300 \times ln 2 \approx 1729 J

Thermodynamic Process Formulas

First Law

Δ U = Q - W, W = \int_{V_{1}}^{V_{2}} P d V

Ideal Gas Law and Temperature

P V = n R T ⟹ T = \frac{P V}{n R}

Process Equations

Process	Constraint	Work Formula
Isothermal	PV = const	W = nRT ln(V₂/V₁)
Adiabatic	PV^γ = const	W = (P₁V₁ − P₂V₂)/(γ−1)
Isobaric	P = const	W = PΔV = P(V₂−V₁)
Isochoric	V = const	W = 0

Heat Capacities (Ideal Gas)

C_{v} = \frac{3}{2} R (monatomic), C_{p} = C_{v} + R, γ = \frac{C _{p}}{C _{v}}

ℹFor a monatomic ideal gas (He, Ar): γ = 5/3 ≈ 1.667. For a diatomic ideal gas (N₂, O₂) at room temperature: γ = 7/5 = 1.4.

Frequently Asked Questions

What does the area under a PV curve represent?

The area under a PV curve equals the work done by the gas. Expansion (V increases) means positive work done by the gas; compression (V decreases) means negative work (work done on the gas). This follows from W = ∫P dV.

Why is the adiabatic curve steeper than the isothermal curve?

During isothermal expansion the temperature stays constant (heat flows in from surroundings), so the pressure follows P = nRT/V. During adiabatic expansion, no heat enters, so the gas also cools as it expands, causing an additional pressure drop beyond the isothermal curve.

What is γ (gamma) and why does it matter?

γ = Cp/Cv is the heat capacity ratio (adiabatic index). It determines how steeply pressure falls during adiabatic expansion. For monatomic gases γ = 5/3; for diatomic gases γ = 7/5. A larger γ means a steeper adiabatic curve.

Why is ΔU = 0 for an isothermal process?

For an ideal gas, internal energy depends only on temperature (U = nCvT). Since temperature is constant in an isothermal process, ΔT = 0 and therefore ΔU = 0. All heat added goes directly into work done by the gas.

How do I read T from the PV diagram?

From the ideal gas law, T = PV/(nR). Any point on the diagram has coordinates (V, P), so you can compute the temperature. Isothermal curves are curves of constant PV; higher isotherms (further from origin) correspond to higher temperatures.