Isothermal Work Calculator

An isothermal process occurs at constant temperature. For an ideal gas, PV = nRT = constant, so the work done equals nRT times the natural log of the volume ratio.

🌡️ Thermodynamics📐 W = nRT ln(V₂/V₁)💨 Gas Laws
Moles of Gas n (mol)
Temperature T (K)
Initial Volume V₁ (m³)
Final Volume V₂ (m³)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Work DoneWW = nRT ln(V2/V1)J
MolesnAmount of gasmol
TemperatureTMust be in KelvinK
Gas ConstantR8.314 J/mol·KJ/mol·K
Pressure VersionWW = P1V1 ln(V2/V1)J

Step-by-Step Examples

Example 1
Isothermal Expansion

1 mol ideal gas at T=300 K expands from V1=0.001 to V2=0.01 m³.

  • W = nRT ln(V2/V1) = 1 × 8.314 × 300 × ln(10)
  • W = 2494 × 2.303 = 5743 J
✓ 5743 J of work done by gas
Example 2
Isothermal Compression

2 mol at 400 K compressed from 0.01 to 0.002 m³.

  • W = 2 × 8.314 × 400 × ln(0.002/0.01)
  • W = 6651 × ln(0.2) = 6651 × (-1.609) = -10,706 J
✓ 10,706 J of work done ON the gas
Example 3
Pressure from Isothermal

P1=100 kPa, V1=0.005 m³, isothermal expansion to V2=0.025 m³.

  • W = P1V1 ln(V2/V1) = 100,000 × 0.005 × ln(5)
  • W = 500 × 1.609 = 804.7 J
  • P2 = P1V1/V2 = 500/0.025 = 20 kPa
✓ 804.7 J work, final pressure = 20 kPa

Real-World Applications

🧊
Refrigeration Cycle
Ideal refrigerator compresses/expands gas through isothermal and adiabatic stages alternately.
⚙️
Carnot Cycle
Two isothermal and two adiabatic stages. Isothermal work determines the heat exchange in Carnot efficiency.
💨
Compressed Gas Systems
Gas expanding from tank to atmosphere approximates isothermal (slow enough for heat exchange).
🧪
Osmotic Systems
Osmotic work against concentration gradients follows log relationship similar to isothermal work.

Common Mistakes to Avoid

⚠️
Using Celsius instead of Kelvin

T in W = nRT ln(V2/V1) must be in Kelvin. 25°C = 298 K.

⚠️
Wrong sign interpretation

W > 0 = gas does work (expansion). W < 0 = work done on gas (compression). Sign convention follows thermodynamics standard.

⚠️
Assuming real gases are ideal

At high pressure or low temperature, real gases deviate from ideal. Van der Waals equation gives correction.

Frequently Asked Questions

Where does the heat go in an isothermal expansion? ▾
Temperature stays constant, so internal energy does not change. The work done by the gas exactly equals heat absorbed: Q = W (first law: dU = 0).
How is isothermal different from adiabatic? ▾
Isothermal: T constant, PV = constant, heat exchanged with surroundings. Adiabatic: Q=0, PV^gamma = constant, temperature changes.
What is a quasi-static process? ▾
A very slow process where each infinitesimal step is in equilibrium. Isothermal processes require quasi-static execution (slow) for heat exchange to maintain T.
How does isothermal relate to Carnot efficiency? ▾
Carnot cycle has two isothermal stages: heat absorption at T_H and rejection at T_C. The net work = Q_H - Q_C = Q_H(1 - T_C/T_H).
Can an isothermal process occur instantaneously? ▾
No. Isothermal requires time for heat exchange. Very fast processes are adiabatic. Real engines operate between these extremes.
What is isothermal compressibility? ▾
kT = -(1/V)(dV/dP)_T, the fractional volume change per unit pressure at constant temperature. For ideal gas: kT = 1/P.
How much heat is absorbed in isothermal expansion? ▾
For ideal gas, Q = W = nRT ln(V2/V1). All work done by gas comes from heat absorbed from surroundings.
What is free expansion? ▾
Adiabatic expansion into vacuum (Q=0, W=0). Temperature stays constant for ideal gas. Entropy increases. This is irreversible.

Related Physics Calculators

🔥 Adiabatic Process⚙️ Carnot Efficiency💨 Ideal Gas Law💨 Specific Heat Ratio💨 Gas Pressure

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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