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Thermal Physics

Maxwell's Law of Velocity Distribution

Maxwell's law of velocity distribution gives the statistical probability distribution of molecular speeds and velocities in an ideal gas in thermal equilibrium. It forms the foundation of the kinetic theory of gases and microscopic thermodynamics.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of Maxwell distribution, expressions for most probable, average, and root-mean-square velocities, and their dependence on temperature and molecular mass.

PGT extension

Detailed derivation of the distribution function using Boltzmann factor and phase space, transformation from velocity coordinates to speed coordinates, and evaluation of mean energy.

Prerequisites

Kinetic theory of gases and ideal gas equation
Concept of temperature and thermal equilibrium
Maxwell-Boltzmann statistics and probability theory

Postulates of Maxwell's Distribution

Molecules of a gas are in continuous random motion, colliding elastically with one another and the walls of the container.
The distribution of velocities is isotropic, meaning no direction is preferred in space.
The velocity components along orthogonal Cartesian axes (x, y, z) are statistically independent events.

The Velocity Distribution Function

The probability dN/N of finding a molecule with velocity components between v_x and v_x + dv_x, etc., is given by the Maxwell distribution function.
It depends explicitly on the mass of the molecule m, absolute temperature T, and Boltzmann constant k_B.
The exponent contains the kinetic energy term divided by thermal energy k_BT.

Speed Distribution Function

Integrating the velocity distribution over all solid angles in velocity space yields the Maxwell speed distribution function f(v).
The function starts at zero for v = 0, rises to a maximum at the most probable speed, and then decreases asymptotically as v approaches infinity.
The area under the speed distribution curve represents the total fraction of molecules, equal to unity.

Characteristic Speeds

Most probable speed (v_mp) is the speed possessed by the maximum fraction of gas molecules.
Mean velocity (v_av) is the arithmetic average of the speeds of all molecules.
Root-mean-square velocity (v_rms) is directly proportional to the square root of temperature and inversely proportional to the square root of molecular mass.

Derivations

Derivation of Maxwell Speed Distribution from Velocity Distribution

Conditions: Isotropic gas in thermal equilibrium

Start with the 3D velocity distribution function: d$n = N (m / 2π k_B T)^{3/2} exp(-m(v_x^2 + v_y^2 + v_z^2) / 2k_B T) dv_x dv_y dv_z$.
Convert Cartesian velocity components to spherical polar coordinates in velocity space: dv_x dv_y d$v_z = v^2 \sin(\theta) dv d\theta dphi$.
Integrate over all angles (\theta from 0 to pi, phi from 0 to 2π) to obtain the number of molecules with speeds between v and v + dv: d$n = 4π N (m / 2π k_B T)^{3/2} v^2 exp(-mv^2 / 2k_B T) dv$.

Result: f(v) d$v = 4π (m / 2π k_B T)^{3/2} v^2 exp(-mv^2 / 2k_B T) dv$

Derivation of Most Probable Speed

Conditions: From Maxwell speed distribution function

Write the speed distribution function f(v) = 4π N (m / 2π k_B T)^{3/2} v^2 exp(-mv^2 / 2k_B T).
Differentiate f(v) with respect to v and set the derivative equal to zero to find the extremum: df(v)/dv = 0.
Solve the resulting algebraic equation for v, discarding the trivial roots at v = 0 and v = infty.

Result: $v_{mp} = \sqrt{\frac{2k_B T}{m}} = \sqrt{\frac{2RT}{M}}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Maxwell Speed Distribution Function	$$dN_v = 4\pi N \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}} dv$$	d$N_v = number of molecules with speeds between v$ and v+dv, N = total molecules, m = mass of molecule, T = temperature Dimensionless fraction or m^{-1}s	Ideal gas in thermal equilibrium
Most Probable Speed	$$v_{mp} = \sqrt{\frac{2k_B T}{m}} = \sqrt{\frac{2RT}{M}}$$	R = universal gas constant, M = molar mass m/s	Maxwellian distribution
Average Speed	$$v_{av} = \sqrt{\frac{8k_B T}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$$	$v_{av} = mean speed$ m/s	Maxwellian distribution
Root Mean Square Speed	$$v_{rms} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3RT}{M}}$$	$v_{rms} = root mean square speed$ m/s	Maxwellian distribution
Hierarchy of Speeds	$$v_{mp} : v_{av} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$$	Ratio constants for characteristic speeds Dimensionless ratio	Same gas and temperature
Average Kinetic Energy per Molecule	$$\langle E \rangle = \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_B T$$	\langle E \rangle = average kinetic energy Joule (J)	Equipartition theorem application
Boltzmann Factor	$$P(E) \propto e^{-\frac{E}{k_B T}}$$	P(E) = relative probability of state with energy E Dimensionless	Canonical ensemble
Fraction of Molecules in Speed Range	$$\frac{dN}{N} = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}} dv$$	dN/N = probability fraction Dimensionless	Equilibrium state

Graphs and experiments

Maxwell Speed Distribution Curve

Draw: A plot of f(v) vs speed v showing an asymmetric bell-shaped curve that rises steeply from zero, peaks at v_mp, and tails off asymptotically towards higher speeds.

Interpret: The peak corresponds to the most probable speed. The area under the curve between any two speeds gives the fraction of molecules having speeds in that range.

Exam use: Directly asked in multiple-choice questions regarding the relative positions of v_mp, v_av, and v_rms on the curve.

Effect of Temperature on Distribution Curve

Draw: Two curves on the same axes: one at temperature T_1 and a broader, lower-peaked curve shifted to the right at a higher temperature T_2 (T_2 > T_1).

Interpret: As temperature increases, the most probable speed increases, the peak shifts to the right, and the curve flattens out to maintain a constant total area under the graph.

Exam use: Tested via qualitative reasoning questions on how temperature changes shift molecular speed distributions.

Effect of Molecular Mass on Distribution Curve

Draw: Curves for two different gases (e.g., Hydrogen and Oxygen) at the same temperature, showing lighter gas having a broader distribution with a higher peak speed.

Interpret: Lighter molecules possess higher velocities on average at the same thermal energy compared to heavier molecules.

Exam use: Questions comparing hydrogen and oxygen gas speed distributions at room temperature.

Solved numericals

TGT core

Calculate the root-mean-square speed of hydrogen molecules at 300 K. Given the molar mass of hydrogen is 2 x 10^{-3} kg/mol and $R = 8.314 J/(mol K)$.

Identify given values: T = 300 K, $M = 2 x 10^{-3} kg/mol$, $R = 8.314 J/(mol K)$.
Use formula: $v_{rms} = \sqrt{3RT / M}$.
Substitute values: $v_{rms} = \sqrt{(3 * 8.314 * 300) / (2 \times 10^{-3})} = \sqrt{7482.6 / 0.002} = \sqrt{3741300}$.
Calculate square root: v_{rms} \approx 1934.2 m/s.

Answer: 1934.2 m/s

PGT extension

At what temperature will the root-mean-square speed of oxygen molecules ($M = 32 g/mol) be equal$ to the root-mean-square speed of hydrogen molecules ($M = 2 g/mol) at -73 °C?$

Find temperature of hydrogen: $T_1 = -73 °C + 273.15 = 200 K$.
Equate rms speeds: \sqrt{3 R T_H / $M_H = \sqrt{3 R T_O / M_O}$.
Simplify relation: T_O / $M_O = T_H / M_H \implies T_O = T_H \times (M_O / M_H)$.
Substitute values: $T_O = 200 \times (32 / 2) = 200 \times 16 = 3200 K$.

Answer: 3200 K

TGT core

If the absolute temperature of an ideal gas is quadrupled, by what factor does the most probable speed of its molecules increase?

Recall the formula for most probable speed: $v_{mp} = \sqrt{2RT / M}$.
Note that v_{mp} is directly proportional to the square root of absolute temperature: v_{mp} \propto \sqrt{T}.
If temperature T becomes 4T, then new speed v'_{mp} = \sqrt{4T} \propto 2\sqrt{T}.
Thus, the speed increases by a factor of 2.

Answer: Factor of 2

Assumptions and limitations

Assumes an ideal gas where intermolecular potentials are negligible except during instantaneous elastic collisions.
Applicable only to gases in thermal equilibrium; fails for systems far from equilibrium or under extremely high densities where quantum degeneracy effects become prominent.

Common misconceptions

Common error: All molecules in a gas at a given temperature travel with the same speed.

Correct: Molecules possess a wide continuous range of speeds distributed statistically according to Maxwell's law.

Collisions constantly redistribute kinetic energy among individual molecules, creating a spread of velocities.

Common error: The most probable speed is equal to the average speed.

Correct: The most probable speed is lower than both the average speed and the root-mean-square speed.

The speed distribution curve is asymmetric and skewed towards higher speeds, causing the mean and rms values to be pulled higher than the peak.

Quick revision

Maxwell distribution gives the statistical distribution of molecular speeds in thermal equilibrium, characterized by v_mp < v_av < v_rms, all scaling as sqrt(T/M).
Maxwell Speed Distribution Function: d$N_v = 4\pi N \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}} dv$
Most Probable Speed: $v_{mp} = \sqrt{\frac{2k_B T}{m}} = \sqrt{\frac{2RT}{M}}$

Must remember

The Maxwell velocity distribution assumes molecules are in continuous random motion, colliding elastically, with an isotropic distribution and statistically independent orthogonal velocity components.
The hierarchy of characteristic speeds is always $v_{mp} : v_{av} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$.
Root-mean-square speed is directly proportional to the square root of absolute temperature $T$ and inversely proportional to the square root of molecular mass $m$ or molar mass $M$.
The Maxwell speed distribution function starts at zero for $v = 0$, reaches a peak at $v_{mp}$, and decreases asymptotically as $v \to \infty$.
Average kinetic energy per molecule depends only on absolute temperature and is given by $\langle E \rangle = \frac{3}{2} k_B T$.

Glossary

Most Probable Speed ($v_{mp}$)The speed possessed by the maximum fraction of gas molecules at a given temperature, given by $\sqrt{\frac{2k_B T}{m}}$.

Mean Velocity ($v_{av}$)The arithmetic average of the speeds of all gas molecules in thermal equilibrium, given by $\sqrt{\frac{8k_B T}{\pi m}}$.

Root-Mean-Square Speed ($v_{rms}$)The square root of the average of the squared speeds of the molecules, given by $\sqrt{\frac{3k_B T}{m}}$.

Boltzmann FactorThe relative probability of a state with energy $E$, proportional to $e^{-\frac{E}{k_B T}}$.

Isotropic DistributionA spatial distribution where no single direction is preferred over any other.

Comparison of Characteristic Speeds

Speed Type	Mathematical Formula	Approximate Numerical Ratio
Most Probable Speed ($v_{mp}$)	$\sqrt{\frac{2k_B T}{m}}$	$1.414$
Average Speed ($v_{av}$)	$\sqrt{\frac{8k_B T}{\pi m}}$	$1.596$
Root-Mean-Square Speed ($v_{rms}$)	$\sqrt{\frac{3k_B T}{m}}$	$1.732$

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following represents the correct hierarchical order of molecular speeds in an ideal gas at absolute temperature $T$?

Explanation: The characteristic speeds follow the ratio $\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$. Hence, $v_{rms} > v_{av} > v_{mp}$.

2. What is the ratio of most probable speed to root-mean-square speed ($v_{mp} : v_{rms}$) for a gas at temperature $T$?

Explanation: Since $v_{mp} = \sqrt{\frac{2k_B T}{m}}$ and $v_{rms} = \sqrt{\frac{3k_B T}{m}}$, their ratio is $\sqrt{2} : \sqrt{3}$.

3. How does the root-mean-square speed of gas molecules change if the absolute temperature of the gas is quadrupled?

Explanation: Since $v_{rms} \propto \sqrt{T}$, increasing the absolute temperature by a factor of 4 increases $v_{rms}$ by a factor of $\sqrt{4} = 2$.

4. The Maxwell speed distribution curve starts at $v = 0$ and ends asymptotically at $v \to \infty$. What is the total area under the curve?

Explanation: The total area under the normalized Maxwell speed distribution curve represents the total fraction of all molecules, which is equal to unity ($1$).

5. Which energy term appears in the exponent of the Maxwell velocity distribution function?

Explanation: The exponent contains the kinetic energy term $\frac{mv^2}{2}$ divided by the thermal energy $k_B T$, forming the Boltzmann factor $e^{-\frac{mv^2}{2k_B T}}$.

6. What is the average kinetic energy per molecule of an ideal gas according to the Maxwell-Boltzmann distribution and equipartition theorem?

Explanation: By the equipartition theorem, each translational degree of freedom contributes $\frac{1}{2} k_B T$. With 3 degrees of freedom, the average kinetic energy per molecule is $\frac{3}{2} k_B T$.

7. If the molar mass of a gas is $M$, what is the expression for its average speed $v_{av}$?

Explanation: The average speed is given by $v_{av} = \sqrt{\frac{8RT}{\pi M}}$, derived by integrating speed over the Maxwell distribution function.

8. Which of the following assumptions is fundamental to Maxwell's distribution of velocities?

Explanation: Postulates of Maxwell's distribution state that velocity components along orthogonal Cartesian axes ($x, y, z$) are statistically independent events.

9. What happens to the most probable speed of gas molecules if the gas is heated from $27^\circ\text{C}$ to $327^\circ\text{C}$?

Explanation: Absolute temperatures are $T_1 = 300\text{ K}$ and $T_2 = 600\text{ K}$. Since $v_{mp} \propto \sqrt{T}$, the speed increases by a factor of $\sqrt{600/300} = \sqrt{2}$.

10. Why is the average speed of gas molecules slightly greater than the most probable speed?

Explanation: The asymmetry of the Maxwell speed distribution curve stretches the tail towards higher speeds, pulling both the average and rms speeds above the peak (most probable speed).

11. What is the probability function $P(E)$ known as in canonical ensembles for energy state $E$?

Explanation: The relative probability of a state with energy $E$ is proportional to $e^{-\frac{E}{k_B T}}$, which is known as the Boltzmann factor.

12. Which factor primarily determines the spread or width of the Maxwell velocity distribution curve at a given temperature?

Explanation: The distribution function depends explicitly on $\frac{m}{k_B T}$, meaning the width and spread scale with temperature and molecular mass.

13. If two different gases, Helium and Hydrogen, are at the same absolute temperature, which gas has a higher root-mean-square speed?

Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, the gas with the smaller molar mass ($M$) has the higher RMS speed. Hydrogen ($M=2$) is lighter than Helium ($M=4$).

14. What is the physical significance of $dN_v$ in the expression $dN_v = 4\pi N \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}} dv$?

Explanation: $dN_v$ represents the differential number of molecules possessing speeds lying in the infinitesimal range between $v$ and $v+dv$.

15. Which velocity component distribution gives an error function profile when integrated in one dimension?

Explanation: The probability distribution for a single velocity component (e.g., $v_x$) is a Gaussian distribution, unlike the speed distribution which includes the $v^2$ volume element factor.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. At what temperature will the root-mean-square speed of oxygen molecules ($M = 32\text{ g/mol}$) be equal to the root-mean-square speed of hydrogen molecules ($M = 2\text{ g/mol}$) at $-73^\circ\text{C}$?

Explanation: $v_{rms} = \sqrt{\frac{3RT}{M}}$. For equal RMS speeds, $\frac{T_{O_2}}{M_{O_2}} = \frac{T_{H_2}}{M_{H_2}}$. $T_{H_2} = 273 - 73 = 200\text{ K}$. $T_{O_2} = 200 \times \frac{32}{2} = 3200\text{ K}$? Wait, calculation: $200 \times 16 = 3200\text{ K}$. In Celsius: $3200 - 273 = 2927^\circ\text{C}$. Let's adjust numbers: If $T_{H_2} = 150\text{ K}$ ($(-123^\circ\text{C})$), $150 \times 16 = 2400\text{ K} = 2127^\circ\text{C}$. Let's revise option values for model question consistency. If $T_{H_2} = 150\text{ K}$, $T_{O_2} = 2400\text{ K}$.

2. The root-mean-square speed of an ideal gas molecule at $300\text{ K}$ is $v_{rms}$. At what temperature will the root-mean-square speed become $2v_{rms}$?

Explanation: Since $v_{rms} \propto \sqrt{T}$, doubling the RMS speed requires increasing the absolute temperature by a factor of $2^2 = 4$. Thus, $T_2 = 4 \times 300\text{ K} = 1200\text{ K}$.

3. If the kinetic energy of translation of 1 mole of gas is $E$, what is the relationship between absolute temperature $T$ and $E$?

Explanation: For 1 mole of gas, total translational kinetic energy is $E = N_A \left(\frac{3}{2} k_B T\right) = \frac{3}{2} R T$, since $R = N_A k_B$.

4. The most probable speed of a gas at temperature $T$ is $400\text{ m/s}$. What is its average speed at the same temperature?

Explanation: Ratio $v_{mp} : v_{av} = \sqrt{2} : \sqrt{\frac{8}{\pi}} = 1 : \sqrt{\frac{4}{\pi}} = 1 : \sqrt{\frac{4}{3.1416}} = 1 : 1.128$. Therefore, $v_{av} = 400 \times 1.128 \approx 451\text{ m/s}$.

5. In the Maxwell speed distribution curve, the peak of the curve shifts in which direction when the temperature of the gas is increased?

Explanation: As temperature increases, $v_{mp}$ shifts to the right (higher speeds), and because the total area under the curve remains constant, the peak height decreases (lower).

6. What is the ratio of root-mean-square speed to the most probable speed ($v_{rms} / v_{mp}$)?

Explanation: $v_{rms} = \sqrt{\frac{3k_B T}{m}}$ and $v_{mp} = \sqrt{\frac{2k_B T}{m}}$. Dividing them gives $\sqrt{\frac{3}{2}}$ or approximately $\sqrt{1.5} \approx 1.224$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which of the following expressions correctly describes the Maxwell speed distribution function $f(v) = \frac{1}{N} \frac{dN}{dv}$?

Explanation: The normalized Maxwell speed distribution function per unit speed interval is $f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}}$.

2. If the temperature of an ideal gas is raised from $27^\circ\text{C}$ to $127^\circ\text{C}$, the percentage change in root-mean-square speed is approximately:

Explanation: $T_1 = 300\text{ K}$, $T_2 = 400\text{ K}$. Ratio $v_2/v_1 = \sqrt{400/300} = \sqrt{4/3} = 1.1547$. Percentage increase = $(1.1547 - 1) \times 100\% \approx 15.5\%$.

3. At what velocity $v$ does the Maxwell speed distribution function $f(v)$ reach its maximum value?

Explanation: The maximum of the speed distribution curve corresponds to the most probable speed $v_{mp} = \sqrt{\frac{2k_B T}{m}}$.

4. Consider a system of gas molecules where velocity components $v_x, v_y, v_z$ are independent. The probability distribution $f(v_x)$ is proportional to:

Explanation: The one-dimensional velocity component distribution along the $x$-axis is given by the Maxwellian Gaussian factor $f(v_x) \propto e^{-\frac{mv_x^2}{2k_B T}}$ without the $v^2$ prefactor.

5. What is the mean translational kinetic energy per degree of freedom for a gas molecule at temperature $T$?

Explanation: According to the equipartition theorem, each independent degree of freedom contributes $\frac{1}{2} k_B T$ to the average energy.

6. If the molar mass of gas A is 4 times that of gas B, and both are at the same temperature, what is the ratio of their most probable speeds $(v_{mp,A} / v_{mp,B})$?

Explanation: $v_{mp} \propto \frac{1}{\sqrt{M}}$. Since $M_A = 4 M_B$, $v_{mp,A} / v_{mp,B} = \sqrt{M_B / M_A} = \sqrt{1/4} = 1/2$.

7. Which of the following statements about the Maxwell speed distribution is correct?

Explanation: The hierarchy of speeds is $v_{rms} (1.732) > v_{av} (1.596) > v_{mp} (1.414)$. Thus, $v_{rms}$ is greater than $v_{av}$.

8. In the derivation of the Maxwell velocity distribution using the Boltzmann factor, why are velocity components treated as independent variables?

Explanation: For an ideal gas, particles are non-interacting except for instantaneous elastic collisions, making total kinetic energy separable into independent orthogonal components ($E = E_x + E_y + E_z$).

9. What is the dimensions or SI unit of the Maxwell speed distribution function $f(v)$?

Explanation: Since $f(v) dv = \frac{dN}{N}$ is dimensionless, $f(v)$ has the units of inverse speed, i.e., $\text{s/m}$ or $(\text{m/s})^{-1}$.

10. Calculate the root-mean-square speed of hydrogen molecules at $0^\circ\text{C}$, given $R = 8.314\text{ J/(mol K)}$ and molar mass $M = 2 \times 10^{-3}\text{ kg/mol}$ roughly.

Explanation: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3 \times 8.314 \times 273}{2 \times 10^{-3}}} = \sqrt{\frac{6811.17}{0.002}} = \sqrt{3405585} \approx 1845\text{ m/s}$. Wait, for $T=273\text{ K}$, $\sqrt{\frac{3 \times 8.314 \times 273}{0.002}} \approx 1840\text{ m/s}$. Let's select $1840\text{ m/s}$ (Option B).