Isothermal Work Ideal Gas Calculator

Formula Used

For a reversible isothermal process of an ideal gas: W = n R T ln(V₂/V₁). Using PV = nRT at constant T, the ratio can also be written as V₂/V₁ = P₁/P₂, so: W = n R T ln(P₁/P₂).

This calculator reports work based on your sign convention selection.

How to Use This Calculator

1. Select an input mode: volumes or pressures.
2. Enter n (moles) and the temperature with units.
3. Enter the initial and final values for your chosen mode.
4. Pick a sign convention that matches your workbook.
5. Click Calculate to view results above the form.
6. Use Download CSV or Download PDF for reports.

Example Data Table

Article: Understanding Isothermal Work in an Ideal Gas

1) What “isothermal” means in practice

An isothermal process holds temperature constant while pressure and volume change. For an ideal gas, constant temperature implies constant internal energy, so the energy transfer is expressed through boundary work and heat flow. In reversible paths, pressure remains well-defined at each step, enabling accurate integration.

2) Work from the reversible pressure–volume curve

Reversible isothermal work comes from the area under the P–V curve. Using P = nRT/V, the integral becomes W = ∫(nRT/V) dV, which evaluates to W = nRT ln(V₂/V₁). This logarithmic dependence makes ratios more important than absolute values.

3) Using pressure ratios when volumes are unknown

At constant temperature, the ideal gas law gives P₁V₁ = P₂V₂. Therefore, V₂/V₁ = P₁/P₂, and the same work can be computed from W = nRT ln(P₁/P₂). This option is helpful when your experiment records only pressures.

4) Sign convention and physical interpretation

Many physics texts take “work done by the gas” as positive during expansion. In that convention, V₂ > V₁ produces a positive logarithm and positive work. If you select “work done on the gas” as positive, the calculator flips the sign to match thermodynamics conventions in some courses.

5) Units and conversions built into the calculator

The calculator converts temperature to kelvin and supports common pressure and volume units. Results are reported in joules, with convenient conversions to kilojoules, megajoules, and watt-hours. When comparing scenarios, keep n in moles and R in joules per mole-kelvin for consistency.

6) What assumptions make the result reliable

The formula applies to ideal-gas behavior and a reversible isothermal path. Real gases may deviate at high pressures or very low temperatures, and rapid processes can be non-reversible. If the process is not quasi-static, measured boundary work can differ from the reversible prediction.

7) Common data issues and how to avoid them

The logarithm requires positive ratios, so inputs must be positive after unit conversion. If V₂ equals V₁ (or P₁ equals P₂), the ratio is one and the work is zero. Verify that initial and final states match the same gas sample and temperature.

8) Reporting results for labs and design notes

For documentation, include the selected input mode, the ratio used, and the sign convention. Export the table to CSV for spreadsheets or generate a PDF for lab notebooks. These outputs help reviewers reproduce the calculation and confirm that the correct reversible isothermal model was applied.

FAQs

1) Does this calculator assume a reversible process?

Yes. The logarithmic work expression comes from integrating along a reversible isothermal path. Irreversible expansions can yield different boundary work even when the initial and final states match.

2) Why does the work depend on a natural logarithm?

For an ideal gas at constant temperature, pressure varies as 1/V. Integrating P dV over that curve produces a natural logarithm of the volume ratio.

3) When should I use the pressure mode?

Use pressure mode when you know P₁ and P₂ more reliably than volumes. At constant temperature, V₂/V₁ = P₁/P₂, so the work remains consistent.

4) What if my temperature is in °C or °F?

Enter the value with the correct unit selector. The calculator converts to kelvin internally before computing work, ensuring compatibility with R in joules per mole-kelvin.

5) Why can my result be negative?

With “work by gas positive,” compression gives V₂/V₁ < 1 and a negative logarithm. If you choose “work on gas positive,” the sign flips to match that convention.

6) Can I change the gas constant R?

Yes. The default is the universal gas constant in SI units. You may adjust R if your coursework specifies a rounded constant, but keep it consistent with kelvin and joules.

7) Is the work equal to heat for isothermal ideal gas?

For an ideal gas, internal energy depends only on temperature. If temperature is constant, ΔU = 0, so heat transfer equals work in magnitude, with signs depending on convention.