Ideal Gas Law Solver

Physics Thermodynamics • Kinetic Theory of Ideal Gases

Written by STEM Calculators Team Published January 3, 2026 Updated February 24, 2026

1. Ideal Gas Law Solver

Solve $$PV=nRT$$ (macroscopic) or $$PV = NkT$$ (molecular). Converts between $$n$$ (moles) and $$N$$ (molecules) via $$N=nN_A$$ . Includes density $\rho=\dfrac{PM}{RT}$ (ideal) and an optional 2nd-virial correction.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Inputs

Solve for: Macro / micro emphasis: Pressure $P$: $P$ unit: Volume $V$: $V$ unit: Temperature $T$: $T$ unit: Moles $n$ (mol): Molecules $N$: Molar mass $M$ (g/mol) for density: Density output:

Tip: You can either enter $n$ or $N$. If both are blank and you are not solving for $n$ or $N$, the solver will error. Use “Solve for” to choose the unknown.

Advanced: partial pressures (mixtures)

Enable partial pressure: Mole fraction $y_i$: Total pressure $P_{\text{tot}}$ (optional): $P_{\text{tot}}$ unit:

Dalton’s law: $P_{\text{tot}}=\sum_i P_i$, and for ideal mixtures $P_i=y_iP_{\text{tot}}$.

University: 2nd virial correction (optional)

Virial correction: Second virial $B$: $B$ unit:

Uses $Z \approx 1 + B/V_m$ with molar volume $V_m=V/n$. Implemented as: $P = \dfrac{nRT}{V}\left(1 + B\dfrac{n}{V}\right)$. (Good at low/moderate densities.)

Graph + kinetic animation

Marker progress $\tau\in[0,1]$: Play speed: PV curve range (around $V$): Shade $\int P\,dV$ (illustrative):

PV graph supports drag-to-pan, wheel-to-zoom, and double-click/Reset view. The kinetic animation shows a small number of particles (visual only); particle speed scales with $\sqrt{T}$.

Ready

Steps

Enter values and click Solve.

Kinetic theory animation (illustrative)

Particles in a box (speed ~ $\sqrt{T}$)

Status: … $v_{\mathrm{rms}}$=…

PV graph (isotherm + state marker)

Isotherm $P(V)$ through the solved state (ideal / optional virial)

Hover: (V,P)=… Marker: τ=1

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Ideal Gas Law Solver — Theory (Kinetic Theory of Ideal Gases)

The ideal gas law connects macroscopic state variables: $$PV=nRT$$ . The same relationship can be written in a molecular form $$PV = NkT$$ , where $N$ is the number of molecules and $k$ is Boltzmann’s constant. The bridge between the two views is Avogadro’s constant $N_A$, where $N=nN_A$, and $R=N_Ak$.

1) Macroscopic form

\[ PV = nRT. \] Here $P$ is pressure, $V$ is volume, $T$ is absolute temperature (K), and $n$ is the amount of gas in moles. The gas constant is \[ R = 8.314462618\ \text{J/mol·K}. \]

2) Molecular form and kinetic-theory connection

\[ PV = NkT,\qquad N = nN_A,\qquad R=N_Ak. \] Constants: \[ k = 1.380649\times10^{-23}\ \text{J/K},\quad N_A = 6.02214076\times10^{23}\ \text{mol}^{-1}. \]

In kinetic theory, pressure arises from molecular collisions with the container walls. Temperature is proportional to average translational kinetic energy: $\langle K \rangle = \tfrac{3}{2}kT$.

3) Density relation

Using $\rho = m/V$ and molar mass $M$ (kg/mol), the ideal-gas density is:

\[ \rho = \frac{PM}{RT}. \]

The calculator accepts $M$ in g/mol and converts internally to kg/mol.

4) Partial pressures (mixtures)

For an ideal mixture (Dalton’s law), total pressure is the sum of partial pressures: $P_{\text{tot}}=\sum_i P_i$, and

\[ P_i = y_i P_{\text{tot}}, \] where $y_i$ is the mole fraction of component $i$.

5) University extension: virial correction

Real gases deviate from ideal behavior. A common low-density correction is the second-virial form:

\[ Z \equiv \frac{PV}{nRT} \approx 1 + \frac{B}{V_m}, \qquad V_m=\frac{V}{n}. \] In the calculator this is implemented as: \[ P = \frac{nRT}{V}\left(1 + B\frac{n}{V}\right). \]

This is an approximation (best at moderate-to-low densities) and depends on temperature through $B(T)$ in real applications.

6) Avogadro fact (sample input)

At $P=1\ \text{atm}$ and $T\approx 273\ \text{K}$, one mole of an ideal gas occupies about $22.4\ \text{L}$ (standard molar volume). So if $P=1\ \text{atm}$, $V=22.4\ \text{L}$, $T=273\ \text{K}$, then $n\approx 1\ \text{mol}$ and $N\approx N_A\approx 6.022\times10^{23}$ molecules.