Isothermal Process Analyzer

Physics Thermodynamics • First Law of Thermodynamics

Written by STEM Calculators Team Published January 2, 2026 Updated February 24, 2026

1. Isothermal Process Analyzer

Analyze an isothermal ideal-gas process (\(\Delta T=0\Rightarrow \Delta U=0\), so \(Q=W\) for the system sign convention). Enter the states, compute work/heat, and visualize the \(P\!-\!V\) hyperbola with shaded work.

Input supports: pi, e, sqrt( ), sin, cos, exp, log. Use * for multiplication. Volumes/pressures are handled via unit selectors.

\(n\) (mol): \(T\) (K) constant:

States

\(V_1\): \(V_2\): Volume unit: Pressure unit (display):

For an ideal gas isothermal process, \(PV=nRT\) is constant. The tool computes \(P_1\) and \(P_2\) from your \(n,T,V_1,V_2\).

Sign convention

Work sign: Entropy preview:

Animation / graph controls

Progress \(\tau\in[0,1]\): Play speed:

The shaded area shows work accumulated from \(V_1\) to the current \(V(\tau)\). Drag to pan, wheel to zoom, double-click or “Reset view” to restore.

Ready

Steps

Enter values, then click Analyze.

PV Graph (isothermal)

\(P(V)=\dfrac{nRT}{V}\) with work area

Hover to probe \((V,P)\).

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Theory — Isothermal Process Analyzer

An isothermal process keeps temperature constant (\(T=\) const). For an ideal gas, internal energy depends only on temperature, so if \(\Delta T=0\) then \(\Delta U=0\). The first law then implies heat and work are equal in magnitude (with the same sign convention).

1) Ideal gas & the isotherm on a \(P\!-\!V\) diagram

From \(PV=nRT\) with constant \(T\):

\[ PV=nRT\quad\Rightarrow\quad P(V)=\frac{nRT}{V} \]

The curve \(P\propto \frac{1}{V}\) is a hyperbola. Moving to larger \(V\) is an expansion; to smaller \(V\) is a compression.

2) First law for an isothermal ideal gas

The first law of thermodynamics is:

\[ \Delta U = Q - W_{\text{by}} \]

For an ideal gas, \(U=U(T)\), so:

\[ \Delta T=0\quad\Rightarrow\quad \Delta U=0\quad\Rightarrow\quad Q=W_{\text{by}} \]

If you use the “work on the system” convention \(W_{\text{on}}=-W_{\text{by}}\), then the matching heat sign changes as well.

3) Reversible isothermal work (maximum work)

For a reversible isothermal process:

\[ W_{\text{by}}=\int_{V_1}^{V_2}P\,dV =\int_{V_1}^{V_2}\frac{nRT}{V}\,dV =nRT\ln\!\left(\frac{V_2}{V_1}\right) \]

For given endpoints \((V_1,T)\to(V_2,T)\), the reversible path gives the maximum magnitude of work (in the appropriate direction).

4) Entropy preview (reversible isothermal)

If the process is reversible:

\[ \Delta S=\frac{Q_{\text{rev}}}{T}=nR\ln\!\left(\frac{V_2}{V_1}\right) \]

5) Reading the shaded area on the graph

On a \(P\!-\!V\) plot, the work done by the gas equals the area under the curve:

\[ W_{\text{by}}=\int_{V_1}^{V_2}P\,dV \]

The calculator shades the area from \(V_1\) to the current animated \(V(\tau)\) to show how work accumulates during the process.

University extension: include real-gas corrections (e.g., van der Waals) and compare deviations from the hyperbola at high pressure / low volume.

Steps

PV Graph (isothermal)

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