The kinetic molecular theory (often called the kinetic theory of gases) explains macroscopic gas behavior by modeling a gas as many tiny particles in constant random motion. Gas pressure, temperature effects, and the ideal gas law can be traced to molecular collisions and the distribution of molecular speeds.
Core postulates of kinetic molecular theory
| Postulate (ideal-gas model) | Physical meaning | Key consequence |
|---|---|---|
| Gas particles are far apart; particle volume is negligible. | Most of the container is empty space. | Volume \(V\) is essentially container volume. |
| Particles move randomly in straight lines between collisions. | No net drift in any direction at equilibrium. | Macroscopic properties emerge from averages over many particles. |
| Collisions (particle–particle and particle–wall) are perfectly elastic. | Total kinetic energy is conserved in collisions. | Pressure arises from momentum transfer, not energy loss. |
| No intermolecular attractions or repulsions (except during collisions). | Potential energy effects are ignored. | Model works best at low pressure and high temperature. |
| Average translational kinetic energy depends only on absolute temperature \(T\). | Temperature measures molecular translational motion. | \(\overline{E_k}=\frac{3}{2}k_BT\) per molecule (ideal gas). |
How molecular collisions create gas pressure
Consider a cubic container of side length \(L\) (volume \(V=L^3\)) containing \(N\) molecules, each of mass \(m\). For a single molecule with velocity component \(v_x\) toward a wall perpendicular to the \(x\)-axis:
- The momentum change on an elastic bounce from that wall is \(\Delta p_x = 2mv_x\).
- The time between successive hits on the same wall is \(\Delta t = \frac{2L}{v_x}\), so the collision frequency with that wall is \(\frac{1}{\Delta t}=\frac{v_x}{2L}\).
- The average force contribution from that molecule to the wall is \[ F_x=\frac{\Delta p_x}{\Delta t}= \frac{2mv_x}{2L/v_x}=\frac{mv_x^2}{L}. \]
Summing over all molecules and dividing by the wall area \(A=L^2\) gives the pressure: \[ P=\frac{F}{A}=\frac{1}{L^2}\sum_{i=1}^{N}\frac{m v_{x,i}^2}{L}=\frac{m}{V}\sum_{i=1}^{N} v_{x,i}^2. \] Because molecular motion is random and isotropic at equilibrium, \[ \overline{v_x^2}=\overline{v_y^2}=\overline{v_z^2}=\frac{1}{3}\overline{c^2}, \] where \(c^2=v_x^2+v_y^2+v_z^2\). Therefore, \[ P=\frac{Nm\overline{v_x^2}}{V}=\frac{1}{3}\frac{Nm\overline{c^2}}{V}. \] Equivalently, \[ PV=\frac{1}{3}Nm\overline{c^2}. \]
Connecting kinetic theory to temperature and the ideal gas law
The average translational kinetic energy per molecule is \[ \overline{E_k}=\frac{1}{2}m\overline{c^2}. \] Combine this with \(PV=\frac{1}{3}Nm\overline{c^2}\) to obtain \[ PV=\frac{2}{3}N\overline{E_k}. \] For an ideal gas, kinetic molecular theory identifies absolute temperature \(T\) as the measure of average translational kinetic energy: \[ \overline{E_k}=\frac{3}{2}k_BT, \] where \(k_B\) is the Boltzmann constant. Substituting gives \[ PV = Nk_BT. \] Since \(N=nN_A\) and \(R=N_Ak_B\), the same result becomes the familiar ideal gas law: \[ PV=nRT. \]
Root-mean-square speed and its temperature dependence
The root-mean-square (rms) speed \(u_{\mathrm{rms}}\) is defined by \(u_{\mathrm{rms}}=\sqrt{\overline{c^2}}\). Using \(\overline{E_k}=\frac{1}{2}m\overline{c^2}=\frac{3}{2}k_BT\), \[ \overline{c^2}=\frac{3k_BT}{m}. \] Converting to molar quantities (\(m = \frac{M}{N_A}\), where \(M\) is molar mass in \(\text{kg/mol}\)) yields \[ u_{\mathrm{rms}}=\sqrt{\overline{c^2}}=\sqrt{\frac{3RT}{M}}. \] The scaling \(u_{\mathrm{rms}}\propto \sqrt{T}\) is a central prediction of kinetic molecular theory.
Visualization: Maxwell–Boltzmann speed distributions at two temperatures
Worked examples
Example 1 (average translational kinetic energy per molecule at room temperature).
For \(T=298\,\text{K}\), \[ \overline{E_k}=\frac{3}{2}k_BT=\frac{3}{2}(1.380649\times 10^{-23}\,\text{J/K})(298\,\text{K}) \approx 6.17\times 10^{-21}\,\text{J}. \] This value is per molecule and refers to translational motion only.
Example 2 (rms speed of nitrogen at \(300\,\text{K}\)).
For \( \mathrm{N_2} \), \(M=0.0280\,\text{kg/mol}\). Using \(R=8.314\,\text{J/(mol·K)}\), \[ u_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}} =\sqrt{\frac{3(8.314)(300)}{0.0280}} \approx 5.17\times 10^{2}\,\text{m/s}. \] Doubling the temperature would increase \(u_{\mathrm{rms}}\) by a factor of \(\sqrt{2}\), not by \(2\).
Common implications and limits of the model
- Pressure increases if \(N\) increases (more collisions per unit time), if \(T\) increases (faster particles transfer more momentum), or if \(V\) decreases (more frequent wall impacts).
- Temperature is not “heat”: in kinetic molecular theory, temperature tracks average translational kinetic energy, not the total energy content of a system.
- Deviations from ideality occur at high pressure and low temperature, where finite particle volume and intermolecular forces become important (real-gas behavior).
Kinetic molecular theory provides the microscopic explanation that links molecular motion to macroscopic gas laws, yielding \(PV=Nk_BT\), \(\overline{E_k}=\frac{3}{2}k_BT\), and \(u_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}\) as central quantitative results for ideal gases.