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Thermal Physics

Average Velocity of Gas Molecules in Kinetic Theory

Average velocity in kinetic theory refers to the arithmetic mean of the velocities of all molecules in a given gas mass at thermal equilibrium. Due to random isotropic motion in a stationary container, its vector sum and average magnitude vanish identically, making scalar speeds like root-mean-square and mean speed physically relevant.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of molecular speeds, distinction between average velocity, mean speed, and root-mean-square speed, and dependence on temperature and molar mass.

PGT extension

Maxwell-Boltzmann velocity distribution function, probability density integration, most probable speed, and relationship between different statistical speeds.

Prerequisites

Concept of Temperature
Kinetic theory of gases
Law of equipartition of energy

Definition and Vector Nature of Average Velocity

The average velocity vector of gas molecules is defined as the arithmetic mean of the individual velocity vectors: $v_av = (1/N) \Sigma v_i$.
In a gas at rest inside a stationary container, molecules move equally in all directions with no net directional flow.
Consequently, the average velocity vector is identically zero: $v_av = 0$.

Distinction Between Average Velocity and Mean Speed

While average velocity considers directional vectors, mean speed (or average speed) considers the scalar magnitudes of velocities.
Mean speed $v_bar = (1/N) \Sigma |v_i|$ is non-zero and always positive.
Mean speed is derived by integrating the Maxwell-Boltzmann speed distribution function from zero to infinity.

Dependence on Temperature and Molecular Mass

Characteristic speeds of gas molecules depend directly on the absolute temperature T.
Characteristic speeds vary inversely with the square root of the molar mass M or molecular mass m.
Heavier gases possess lower average speeds compared to lighter gases at the same thermodynamic temperature.

Maxwell-Boltzmann Speed Distribution Overview

The distribution gives the fraction of molecules having speeds in the range between v and v + dv.
It incorporates three characteristic speeds: most probable speed (v_mp), mean speed (v_bar), and root-mean-square speed (v_rms).
The hierarchy of these speeds is always v_mp < v_bar < v_rms for any ideal gas at absolute temperature T.

Derivations

Derivation of Mean Speed from Maxwell-Boltzmann Distribution

Conditions: Ideal gas in thermal equilibrium obeying Maxwell-Boltzmann statistics

Write the Maxwell-Boltzmann distribution function for molecular speeds: dn/$n = 4\pi (m / 2\pikT)^(3/2) v^2 exp(-mv^2 / 2kT) dv$.
Set up the integral for mean speed: $v_bar = \int v (dn/n) from v = 0$ to $v = \infty $.
Evaluate the definite standard integral: ∫ v^3 exp(-mv^2 / 2kT) d$v = (2kT/m)^2$.
Substitute the evaluated integral back into the expression and simplify constants.

Result: $v_bar = \sqrt{\frac{8RT}{\pi M}} = \sqrt{\frac{8kT}{\pi m}}$

Derivation of Root-Mean-Square Speed

Conditions: Ideal gas obeying classical kinetic theory assumptions

Start from the kinetic pressure equation of an ideal gas: $P = (1/3) \rho v_{rms}^2$.
Substitute the ideal gas law in terms of density: $P = (\rhoRT) / M$.
Equate the two expressions for pressure: (1/3) ρ v_{rms}^2 = (\rhoRT) / M$.
Solve algebraically for v_{rms}.

Result: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Average Velocity Vector	$$v_{av} = 0$$	$v_{av} = vector average velocity of gas molecules$ m s^{-1}	Isotropic gas in a stationary enclosure
Mean Speed	$$v_{bar} = \sqrt{\frac{8RT}{\pi M}}$$	R = universal gas constant, T = absolute temperature, M = molar mass m s^{-1}	Maxwellian distribution at temperature T
Root-Mean-Square Speed	$$v_{rms} = \sqrt{\frac{3RT}{M}}$$	R = universal gas constant, T = absolute temperature, M = molar mass m s^{-1}	Ideal gas
Most Probable Speed	$$v_{mp} = \sqrt{\frac{2RT}{M}}$$	R = universal gas constant, T = absolute temperature, M = molar mass m s^{-1}	Peak of Maxwell-Boltzmann distribution curve
Ratio of Speeds	$$v_{mp} : v_{bar} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1 : 1.128 : 1.225$$	Comparison of statistical speeds Dimensionless ratio	Any ideal gas at constant temperature
Temperature Dependence	$$v \propto \sqrt{T}$$	v = any characteristic molecular speed, T = absolute temperature Dimensionless proportionality	Constant molar mass
Molar Mass Dependence	$$v \propto \frac{1}{\sqrt{M}}$$	v = any characteristic molecular speed, M = molar mass Dimensionless proportionality	Constant absolute temperature
Kinetic Energy per Molecule via RMS speed	$$E = \frac{1}{2} m v_{rms}^2 = \frac{3}{2} k T$$	m = mass of one molecule, k = Boltzmann constant, T = absolute temperature J	Monatomic ideal gas

Graphs and experiments

Maxwell-Boltzmann Speed Distribution Curve

Draw: Asymmetric bell-shaped curve of fraction of molecules versus molecular speed, marking v_mp, v_bar, and v_{rms}.

Interpret: Shows that very slow and very high speeds are rare, with the peak representing the most probable speed.

Exam use: Tested in PGT multiple-choice questions regarding the relative positions of v_mp, v_bar, and v_{rms}.

Effect of Temperature on Speed Distribution

Draw: Two curves on the same axes: one at lower temperature T_1 (taller peak, shifted left) and higher temperature T_2 (broader, flatter peak, shifted right).

Interpret: Higher temperature increases the spread of speeds and shifts the peak to a higher velocity value while maintaining equal total area under the curve.

Exam use: Direct conceptual questions on how heating alters molecular speed distribution.

Effect of Molar Mass on Speed Distribution

Draw: Curves for a heavy gas (like CO_2) and a light gas (like H_2) at the same temperature.

Interpret: Lighter gas curve peaks at a higher speed with a wider spread; heavier gas curve is sharper and shifted toward lower speeds.

Exam use: Comparative analysis questions in TGT and PGT examinations.

Solved numericals

TGT core

Calculate the root-mean-square speed of hydrogen molecules at 300 K. Take $R = 8.314 J mol^{-1} K^{-1}$ and $M = 2 x 10^{-3} kg mol^{-1}$.

Identify the formula: $v_{rms} = \sqrt{\frac{3RT}{M}}$.
Substitute values: R = 8.314, T = 300, $M = 2 x 10^{-3}$.
Compute value: (3 * 8.314 * 300) / (2 x 10^{-3}) = 3,741,300.
Take square root: \sqrt{3,741,300} = 1934.25 m s^{-1}.

Answer: 1934.25 m s^{-1}

PGT extension

At what temperature will the root-mean-square speed of oxygen molecules ($M = 32 g mol^{-1}) be equal$ to the root-mean-square speed of helium molecules ($M = 4 g mol^{-1}) at -73 °C?$

Convert helium temperature to Kelvin: $T_{He} = -73 + 273 = 200 K$.
Equate rms speeds: \sqrt{\frac{3RT_{O2}}{M_{O2}}} = \sqrt{\frac{3RT_{He}}{M_{He}}}.
Simplify relation: \frac{T_{O2}}{M_{O2}} = \frac{T_{He}}{M_{He}}.
Solve for T_{O2}: $T_{O2} = T_{He} \times (M_{O2} / M_{He}) = 200 \times (32 / 4) = 200 \times 8 = 1600 K$.

Answer: 1600 K

PGT extension

For a gas obeying Maxwell-Boltzmann distribution, find the ratio of the mean speed to the root-mean-square speed.

Recall expression for mean speed: $v_{bar} = \sqrt{\frac{8RT}{\pi M}}$.
Recall expression for root-mean-square speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$.
Form the ratio: \frac{v_{bar}}{v_{rms}} = \frac{\sqrt{8 / \pi}}{\sqrt{3}} = \sqrt{\frac{8}{3\pi}}.
Evaluate numerically: \sqrt{8 / (3 * 3.14159)} = \sqrt{0.8488} \approx 0.9213.

Answer: \sqrt{\frac{8}{3\pi}} \approx 0.921

Assumptions and limitations

Molecules are treated as point masses interacting solely via elastic collisions, ignoring intermolecular forces except during collisions.
The gas is dilute and behaves ideally, obeying PV = nRT.
Maxwell-Boltzmann distribution assumes thermodynamic equilibrium and a large number of molecules, failing at extremely low temperatures or high quantum densities.

Common misconceptions

Common error: The average velocity of gas molecules is equal to the mean speed.

Correct: Average velocity is zero for a stationary gas, whereas mean speed is a positive scalar quantity.

Velocity is a vector quantity, and random motion in all directions causes vector cancellation.

Common error: All molecules in a gas at temperature T move at the same speed.

Correct: Molecules possess a wide continuous range of speeds governed by the Maxwell-Boltzmann distribution.

Thermal collisions constantly redistribute kinetic energy among individual molecules.

Common error: Root-mean-square speed is the arithmetic average of molecular speeds.

Correct: RMS speed is the square root of the average of squared speeds, giving higher weight to faster molecules.

The mathematical definition involves squaring speeds before averaging, unlike the simple arithmetic mean speed.

Quick revision

Average velocity of a confined gas is always zero due to random isotropic motion.
Mean speed is given by \sqrt{8RT / \pi M}.
Root-mean-square speed is given by \sqrt{3RT / M}.
Most probable speed is given by \sqrt{2RT / M}.
All characteristic molecular speeds are directly proportional to the square root of absolute temperature and inversely proportional to the square root of molar mass.

Must remember

The vector average velocity of gas molecules in a stationary container is identically zero ($v_{av} = 0$) due to isotropic random motion.
Mean speed ($v_{bar}$), root-mean-square speed ($v_{rms}$), and most probable speed ($v_{mp}$) are all non-zero scalar quantities.
The hierarchy of characteristic speeds is always $v_{mp} < v_{bar} < v_{rms}$.
Characteristic molecular speeds vary directly with the square root of absolute temperature ($v \propto \sqrt{T}$) and inversely with the square root of molar mass ($v \propto 1/\sqrt{M}$).

Glossary

Average Velocity Vector ($v_{av}$)The arithmetic mean of individual molecular velocity vectors, which equals zero for a gas at rest.

Mean Speed ($v_{bar}$)The arithmetic mean of the scalar magnitudes of molecular speeds, given by $\sqrt{\frac{8RT}{\pi M}}$.

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

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

Average Velocity vs Mean Speed

Feature	Average Velocity ($v_{av}$)	Mean Speed ($v_{bar}$)
Nature	Vector quantity	Scalar quantity
Value for stationary gas	Always zero ($0$)	Non-zero and positive
Mathematical definition	$\frac{1}{N} \sum \vec{v}_i$	$\frac{1}{N} \sum \|\vec{v}_i\|$

Practice Quiz

Choose an option to reveal the explanation.

1. What is the vector average velocity of gas molecules in a stationary container?

Explanation: Because molecules move equally in all directions in a stationary container, the vector sum of velocities cancels out, making the average velocity vector identically zero ($v_{av} = 0$).

2. Which of the following relations correctly represents the hierarchy of molecular speeds?

Explanation: For any ideal gas at an absolute temperature $T$, the characteristic speeds follow the strict order: most probable speed ($v_{mp}$) < mean speed ($v_{bar}$) < root-mean-square speed ($v_{rms}$).

3. How does the root-mean-square speed of gas molecules depend on the absolute temperature $T$?

Explanation: From the formula $v_{rms} = \sqrt{\frac{3RT}{M}}$, the root-mean-square speed is directly proportional to the square root of the absolute temperature ($v_{rms} \propto \sqrt{T}$).

4. If the molar mass of a gas is quadrupled while keeping the temperature constant, what happens to its mean speed?

Explanation: Mean speed is inversely proportional to the square root of molar mass ($v_{bar} \propto \frac{1}{\sqrt{M}}$). If $M$ is multiplied by $4$, $v_{bar}$ is multiplied by $\frac{1}{\sqrt{4}} = \frac{1}{2}$, meaning it is halved.

5. What is the ratio of most probable speed, mean speed, and root-mean-square speed ($v_{mp} : v_{bar} : v_{rms}$)?

Explanation: The ratio is $\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1.414 : 1.596 : 1.732$, which simplifies relative to $v_{mp}$ to approximately $1 : 1.128 : 1.225$.

6. Which of the following quantities is a scalar and always positive for a gas?

Explanation: Mean speed considers the scalar magnitude of molecular velocities and is always positive, unlike the vector average velocity which can be zero.

7. Which speed corresponds to the peak of the Maxwell-Boltzmann speed distribution curve?

Explanation: The most probable speed ($v_{mp} = \sqrt{\frac{2RT}{M}}$) corresponds to the maximum of the Maxwell-Boltzmann speed distribution curve, representing the speed possessed by the largest fraction of molecules.

8. If absolute temperature $T$ of an ideal gas is increased by a factor of $4$, the root-mean-square speed becomes:

Explanation: Since $v_{rms} \propto \sqrt{T}$, increasing temperature by a factor of $4$ scales the velocity by $\sqrt{4} = 2$ (doubled).

9. Why is the average velocity of gas molecules zero while their root-mean-square speed is non-zero?

Explanation: Velocity is a directional vector quantity leading to cancellation in random thermal motion, whereas RMS speed uses squared scalar magnitudes, ensuring a positive result.

10. What is the expression for kinetic energy of a single molecule in terms of its root-mean-square speed?

Explanation: By definition, the kinetic energy of a single molecule of mass $m$ is given by $E = \frac{1}{2} m v_{rms}^2$.

11. At a given temperature $T$, compare the root-mean-square speed of Hydrogen gas ($H_2$) and Oxygen gas ($O_2$).

Explanation: The molar mass of $O_2$ is $32$ g/mol and $H_2$ is $2$ g/mol. Since $v_{rms} \propto \frac{1}{\sqrt{M}}$, the ratio is $\sqrt{32/2} = \sqrt{16} = 4$. Thus, $v_{rms}(H_2) = 4 v_{rms}(O_2)$.

12. Which of the following statements regarding the Maxwell-Boltzmann distribution is correct?

Explanation: The Maxwell-Boltzmann distribution gives the exact fraction of molecules having speeds in the range between $v$ and $v+dv$ at a given absolute temperature.

13. What is the unit of the root-mean-square speed of gas molecules?

Explanation: Root-mean-square speed is a speed, therefore its SI unit is meters per second (m s$^{-1}$).

14. In the kinetic theory of gases, what does the term $R$ represent in the formula for $v_{rms}$?

Explanation: In $v_{rms} = \sqrt{\frac{3RT}{M}}$, $R$ denotes the universal gas constant.

15. Which molecular speed gives higher statistical weight to faster molecules?

Explanation: Because root-mean-square speed squares the individual speeds before averaging and taking the square root, it assigns a higher statistical weight to faster molecules compared to the simple arithmetic mean speed.

PYQ Patterns

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

1. The root-mean-square speed of the molecules of a gas is proportional to:

Explanation: From the formula $v_{rms} = \sqrt{\frac{3RT}{M}}$, $v_{rms}$ is directly proportional to the square root of the absolute temperature $\sqrt{T}$.

2. If the temperature of an ideal gas is raised from $27^{\circ}\text{C}$ to $927^{\circ}\text{C}$, the root-mean-square speed of its molecules becomes:

Explanation: Absolute temperatures are $T_1 = 27 + 273 = 300$ K and $T_2 = 927 + 273 = 1200$ K. The ratio of speeds is $\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{1200}{300}} = \sqrt{4} = 2$. Thus, the speed is doubled.

3. The ratio of the root-mean-square speed to the mean speed of gas molecules at temperature $T$ is:

Explanation: Root-mean-square speed $v_{rms} = \sqrt{\frac{3RT}{M}}$ and mean speed $v_{bar} = \sqrt{\frac{8RT}{\pi M}}$. Their ratio $\frac{v_{rms}}{v_{bar}} = \sqrt{\frac{3}{\frac{8}{\pi}}} = \sqrt{\frac{3\pi}{8}}$.

4. What is the average velocity vector of an enclosed gas at rest?

Explanation: Due to random isotropic motion in all directions inside a stationary container, the individual velocity vectors cancel out, making the vector average velocity identically zero.

5. At what temperature will the root-mean-square speed of hydrogen molecules be equal to that of oxygen molecules at $300$ K?

Explanation: Equating $v_{rms}$ for hydrogen ($M_1 = 2$) and oxygen ($M_2 = 32$ at $T_2 = 300$ K): $\sqrt{\frac{3RT_1}{2}} = \sqrt{\frac{3R(300)}{32}} \implies \frac{T_1}{2} = \frac{300}{32} \implies T_1 = \frac{600}{32} = 18.75$ K $\approx 19$ K.

6. Which of the following represents the correct order of speeds $v_{mp}$, $v_{bar}$, and $v_{rms}$?

Explanation: The statistical order of speeds derived from the Maxwell-Boltzmann distribution is always $v_{rms} > v_{bar} > v_{mp}$.

Timed Topic Test

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

1. Which of the following statements is true regarding the average velocity and mean speed of gas molecules?

Explanation: For a gas in a stationary container, random motion causes vector cancellation resulting in zero average velocity, whereas mean speed is the sum of scalar magnitudes and is non-zero.

2. If the absolute temperature of a gas is doubled and its molar mass is halved, the root-mean-square speed will increase by a factor of:

Explanation: Since $v_{rms} \propto \sqrt{\frac{T}{M}}$, replacing $T$ with $2T$ and $M$ with $M/2$ gives $\sqrt{\frac{2T}{M/2}} = \sqrt{4 \frac{T}{M}} = 2 \sqrt{\frac{T}{M}}$. Thus, speed increases by a factor of $2$.

3. The most probable speed of gas molecules at absolute temperature $T$ is given by:

Explanation: The most probable speed $v_{mp}$ is derived by finding the peak of the Maxwell-Boltzmann distribution, yielding $v_{mp} = \sqrt{\frac{2RT}{M}}$.

4. Why is root-mean-square speed used in determining the kinetic pressure of gases rather than mean speed?

Explanation: Molecular collisions and momentum transfer rate depend on the mean of the squared velocities ($\bar{v^2}$), which directly connects pressure to the root-mean-square speed.

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

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

6. If the root-mean-square speed of an ideal gas at $300$ K is $v$, at what temperature will the root-mean-square speed be $2v$?

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

7. Which of the following correctly contrasts mean speed and root-mean-square speed?

Explanation: RMS speed involves squaring speeds, averaging, and taking the square root ($\sqrt{\overline{v^2}}$), which mathematically weighs higher speeds more heavily than the linear arithmetic mean speed.

8. What does the Maxwell-Boltzmann distribution curve represent when plotted as fraction of molecules versus speed?

Explanation: The Maxwell-Boltzmann distribution curve is asymmetric (skewed to the right with a long tail towards higher speeds), meaning the mean and RMS speeds are slightly higher than the most probable speed at the peak.

9. If two different gases have the same absolute temperature, which gas molecules will have the higher root-mean-square speed?

Explanation: Since $v_{rms} \propto \frac{1}{\sqrt{M}}$, gases with smaller molar masses (lighter gases) possess higher root-mean-square speeds at the same temperature.

10. Which of the following is the correct dimensionally consistent formula for mean speed?

Explanation: The rigorous derivation from the Maxwell-Boltzmann speed distribution function yields the mean speed formula $v_{bar} = \sqrt{\frac{8RT}{\pi M}}$.