Proportional Patterns in Kinetic Theory
Gas Laws and Properties
| Relationship / Law | Proportional Pattern | Variables Involved | Conditions / Context |
| Boyle’s Law | Pressure is inversely proportional to Volume. | Pressure (P), Volume (V) | Constant Temperature and mass. |
| Charles’ Law | Volume is directly proportional to absolute temperature. | Volume (V), Temperature (T) | Constant Pressure and mass. |
| Pressure & Density | Pressure is directly proportional to number density. | Pressure (P), Number Density (n) | Constant Temperature. |
| Avogadro’s Law | The number of molecules is directly proportional to the volume. | Number of Molecules (N), Volume (V) | Constant Temperature and Pressure. |
| Dalton’s Law | The total pressure is the sum of the partial pressures. | Total Pressure (P_total), Partial Pressures (P₁, P₂, …) | Applies to a mixture of non-reacting ideal gases. |
Kinetic Theory & Molecular Properties
| Relationship / Concept | Proportional Pattern | Variables Involved | Conditions / Context |
| Kinetic Energy & Temp. | Average kinetic energy is directly proportional to absolute temperature. | Average Kinetic Energy (E), Temperature (T) | This is a fundamental concept in kinetic theory. |
| Internal Energy & Temp. | The internal energy of an ideal gas is directly proportional to its absolute temperature. | Internal Energy (E), Temperature (T) | For an ideal gas. |
| Molecular Speed & Temp. | The root-mean-square (rms) speed is proportional to the square root of the absolute temperature. | RMS Speed (v_rms), Temperature (T) | At a fixed molecular mass. |
| Molecular Speed & Mass | The root-mean-square (rms) speed is inversely proportional to the square root of the molecular mass. | RMS Speed (v_rms), Molecular Mass (m) | At a fixed temperature. |
| Gas Diffusion | The rate of diffusion is inversely proportional to the square root of the molar mass. | Rate of Diffusion, Molar Mass (M) | Comparing different gases at the same conditions. |
| Mean Free Path & Density | The mean free path is inversely proportional to the number density of the gas. | Mean Free Path (l), Number Density (n) | Denser gases lead to more frequent collisions. |
| Mean Free Path & Size | The mean free path is inversely proportional to the square of the molecular diameter. | Mean Free Path (l), Molecular Diameter (d) | Larger molecules have a greater chance of colliding. |
Energy Distribution
| Relationship / Law | Proportional Pattern | Variables Involved | Conditions / Context |
| Equipartition of Energy | Energy is distributed equally among all available degrees of freedom. | Total Energy, Degrees of Freedom | A system in thermal equilibrium. |
proportionality patterns found in derived variations of the formulas :
Derived Proportional Patterns in Kinetic Theory
| Relationship / Concept | Proportional Pattern | Variables Involved | Formula / Concept Used |
| Boyle’s Law | Pressure is inversely proportional to Volume. | Pressure (P), Volume (V) | PV = constant (at constant T, N) |
| Charles’ Law | Volume is directly proportional to absolute temperature. | Volume (V), Temperature (T) | V/T = constant (at constant P, N) |
| Pressure and Temperature | Pressure is directly proportional to absolute temperature. | Pressure (P), Temperature (T) | P/T = constant (at constant V, N) from PV = N k_B T |
| Pressure and Density | Pressure is directly proportional to the number density of molecules. | Pressure (P), Number Density (n) | P = n k_B T |
| Kinetic Energy and Temp. | The average translational kinetic energy of a molecule is directly proportional to the absolute temperature. | Avg. Kinetic Energy (E), Temp. (T) | E/N = (3/2) k_B T |
| Internal Energy and Temp. | The internal energy of an ideal gas is directly proportional to its absolute temperature. | Internal Energy (U), Temp. (T) | U = (f/2) RT (where f is degrees of freedom) |
| RMS Speed and Temperature | The root-mean-square (rms) speed is directly proportional to the square root of the absolute temperature. | RMS Speed (v_rms), Temp. (T) | v_rms = √(3 k_B T / m) |
| RMS Speed and Mass | The root-mean-square (rms) speed is inversely proportional to the square root of the molecular mass. | RMS Speed (v_rms), Mass (m) | v_rms = √(3 k_B T / m) |
| Rate of Gas Diffusion | The rate of diffusion is inversely proportional to the square root of the molecular mass. | Rate of Diffusion, Mass (m) | Explained by v_rms ∝ 1/√m. Lighter gases diffuse faster. |
| Pressure and Molecular Speed | Pressure is directly proportional to the mean square speed of the molecules. | Pressure (P), Mean Square Speed (v²) | P = (1/3) n m v² |
| Mean Free Path & Density | The mean free path of a molecule is inversely proportional to the number density of the gas. | Mean Free Path (l), Number Density (n) | l = 1 / (√2 n π d²) |
| Mean Free Path & Molecular Size | The mean free path of a molecule is inversely proportional to the square of its diameter. | Mean Free Path (l), Diameter (d) | l = 1 / (√2 n π d²) |
| Collision Time & Density | The average time between collisions is inversely proportional to the number density of the gas. | Collision Time (τ), Number Density (n) | Derived from τ = l / <v> and l ∝ 1/n |
| Mean Free Path & Temp. | The mean free path is directly proportional to the absolute temperature. | Mean Free Path (l), Temp. (T) | At constant pressure, n ∝ 1/T, so l ∝ 1/n implies l ∝ T. |