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Isothermal Process Calculator

Calculate work W and heat Q for isothermal ideal gas (ΔU = 0)

Parameters

molⓘ
Kⓘ
m³ⓘ
m³ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Work W:
1728.85;J1728.85;J
Heat Q:
1728.85;J1728.85;J
P₁:
103925.00;Pa103925.00;Pa
P₂:
51962.50;Pa51962.50;Pa

Examples

1 mol at 300 K, double volume

V₁=0.024 m³, V₂=0.048 m³.

    Compression

    Halve volume.

      Visualization

      Isothermal Processes — Constant Temperature

      An isothermal process maintains constant temperature throughout. For an ideal gas, internal energy depends only on T (U = U(T)), so ΔU = 0 and the first law gives Q = W: all heat absorbed is converted to work (expansion) or all work done on the gas appears as heat rejected (compression).

      Along an isotherm, the ideal gas law gives PV = nRT = constant — a hyperbola on the P–V diagram. Pressure falls inversely with volume: P₂/P₁ = V₁/V₂. Doubling volume at fixed T halves pressure.

      Reversible isothermal work: W = ∫P dV = ∫(nRT/V) dV = nRT ln(V₂/V₁) = nRT ln(P₁/P₂). Use R = 8.314 J/(mol·K), T in kelvin, and V₂/V₁ dimensionless. For 1 mol at 300 K expanding from 24 L to 48 L, W ≈ 8.314 × 300 × ln(2) ≈ 1730 J.

      Maintaining constant T requires slow contact with a thermal reservoir (heat bath) so heat can flow in during expansion or out during compression. Fast expansion without heat input is adiabatic, not isothermal.

      Isothermal expansion extracts maximum work between two volumes at fixed T compared to adiabatic expansion to the same V₂ (which ends at lower P and T). Carnot cycle uses two isothermal legs at T_hot and T_cold connected by adiabatic steps.

      Real gases deviate slightly: intermolecular forces can make ΔU ≠ 0 even at fixed T. Van der Waals and other equations of state give curved isotherms near the critical point.

      Key Concepts

      • T constant; ideal gas: ΔU = 0
      • First law: Q = W (heat ↔ work exchange)
      • PV = nRT = const — hyperbola on P–V plot
      • W_rev = nRT ln(V₂/V₁) = nRT ln(P₁/P₂)
      • Requires thermal reservoir for slow process
      • Carnot isotherms at T_hot and T_cold

      Real-World Applications

      • Idealized lab gas expansion with water bath thermostat
      • Carnot engine isothermal heat absorption/rejection steps
      • Phase equilibrium at fixed T (vapor pressure, boiling)
      • Maximum reversible work calculations in thermodynamics courses
      • Class 12 derivation of isothermal work integral
      • Industrial gas compression with intercoolers (approximate isothermal stages)

      Explore Further

      More thermodynamics tools

      Physics Equations

      Isothermal Work:
      W=nRTln⁡V2V1W = nRT \ln\frac{V_2}{V_1}
      First Law:
      Q=W(ΔU=0)Q = W \quad (\Delta U = 0)

      Step-by-Step Solution

      See how the main results are calculated.

      1

      Step 1: Isothermal Condition

      Temperature constant → ΔU = 0 for ideal gas.

      Equation:

      ΔU=0⇒Q=W\Delta U = 0 \Rightarrow Q = W

      Explanation:

      All heat converts to work or vice versa.

      2

      Step 2: Ideal Gas at Fixed T

      Equation:

      PV=nRT=constPV = nRT = \text{const}

      Explanation:

      Pressure falls as volume rises on expansion.

      3

      Step 3: Work Formula

      Equation:

      W=nRTln⁡V2V1W = nRT \ln\frac{V_2}{V_1}

      Calculation:

      W=1×8.314×300×ln⁡0.0480.024W = 1 \times 8.314 \times 300 \times \ln\frac{0.048}{0.024}

      Explanation:

      Natural log of volume ratio; V₂ > V₁ → positive work by gas.

      4

      Step 4: Numerical Work

      Calculation:

      W=2494.20×ln⁡(2)=1728.8477 JW = 2494.20 \times \ln(2) = 1728.8477 \text{ J}

      Result:

      W=1728.8477JW = 1728.8477 J
      5

      Step 5: Heat

      Equation:

      Q=WQ = W

      Calculation:

      Q=1728.8477 JQ = 1728.8477 \text{ J}

      Explanation:

      Heat absorbed equals work done in isothermal expansion.

      6

      Step 6: PV Diagram

      Path is hyperbola on P–V plot.

      Explanation:

      Area under curve equals work ∫P dV.

      Frequently Asked Questions (FAQ)

      Real gases at isothermal?

      ΔU not exactly zero if intermolecular forces matter; ideal gas approximation.

      Why ln in work formula?

      From W = ∫P dV with P = nRT/V.

      Isothermal vs isentropic?

      Isentropic: reversible adiabatic (Q=0). Isothermal: T fixed (Q≠0).

      Units for W?

      Joules if nRT in J (use R=8.314, T in K, V ratio dimensionless).

      Maximum work between two reservoirs?

      Carnot uses isothermal steps at T_hot and T_cold.

      Practice MCQs

      1. Isothermal expansion of ideal gas:
      2. In isothermal process Q equals:
      3. Work in expansion V₂ > V₁:
      4. PV curve for isotherm is:
      5. To maintain isothermal expansion you need:
      6. Doubling volume isothermally multiplies pressure by:

      Related Calculators

      These tools connect to the same physics concepts used in this calculator.

      First Law of ThermodynamicsCalculate internal energy change ΔU = Q − W from heat and work.Adiabatic ProcessFind P₂ and T₂ for reversible adiabatic ideal gas expansion (Q = 0).Boyle's LawCalculate pressure or volume at constant temperature (P₁V₁ = P₂V₂).Charles's LawCalculate volume change with temperature at constant pressure.Error PropagationCombine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.Related Measurement & Uncertainty topic.Maxwell-Boltzmann DistributionCalculate velocity distributions and characteristic velocities for ideal gases using Maxwell-Boltzmann statistics.Related Statistical Physics topic.Projectile MotionCalculate and visualize the path of objects under the influence of gravity.Related Mechanics topic.