Home › Physics › Thermal Physics › Kinetic Theory of Gases
Thermal Physics
Kinetic Theory of Gases
The kinetic theory of gases models macroscopic thermodynamic properties of gases by considering microscopic random translational motion of constituent molecules. It establishes the mechanical origin of pressure and temperature in terms of molecular collisions.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Postulates of kinetic theory, pressure of an ideal gas, root-mean-square, average, and most probable speeds, concept of temperature, degrees of freedom, and law of equipartition of energy.
PGT extension
Maxwell-Boltzmann velocity distribution function, mean free path, collision frequency, transport phenomena (viscosity, thermal conduction, diffusion), and real gas corrections via Van der Waals equation.
Prerequisites
Basic laws of thermodynamics
Concept of mole and Avogadro constant
Vector resolution and mechanics of elastic collisions
Postulates of the Kinetic Theory of Gases
A gas consists of a very large number of identical, rigid, spherical molecules moving in random directions with all possible speeds.
The actual volume occupied by the gas molecules is negligible compared to the total volume of the container.
Molecules undergo perfectly elastic collisions with each other and with the walls of the container, conserving both total momentum and kinetic energy.
Pressure of an Ideal Gas
Gas pressure arises due to the continuous bombardment of molecules against the walls of the vessel, resulting in a change of momentum.
The expression for pressure is $P = (1/3)mnv_{rms}^{2}$, where m is mass per molecule, n is numerical density, and vrms is root mean square velocity.
Pressure depends only on the mass density and the mean square speed of the gas molecules, independent of the container shape.
Degrees of Freedom and Equipartition of Energy
Degrees of freedom (f) represent the independent coordinates required to specify the dynamical state of a mechanical system.
Monoatomic, diatomic, and polyatomic gases possess 3, 5, and 6 degrees of freedom respectively at room temperature, with vibrational modes becoming active at high temperatures.
The Law of Equipartition of Energy states that in thermal equilibrium, the total kinetic energy of a dynamical system is distributed equally among all active degrees of freedom, each contributing (1/2)kBT per molecule.
Maxwell-Boltzmann Velocity Distribution
The distribution function provides the probability fraction of molecules having speeds between v and v + dv at temperature T.
It yields three characteristic speeds: most probable speed (vmp), average speed (vavg), and root-mean-square speed (vrms).
The relationship between these characteristic speeds is always vmp < vavg < vrms.
Derivations
Pressure Exerted by an Ideal Gas
Conditions: N identical molecules of mass m inside a cubical container of side L.
Consider a molecule moving with velocity component vx colliding elastically with a wall perpendicular to the x-axis at x = L.
The change in momentum during one collision with the wall is Δpx = -mvx - (mvx) = -2mvx.
The time interval between successive collisions with the same wall is Δ$t = 2L / v_{x}$, giving a collision force Fx = (2mvx) / (2L / vx) = mvx2 / L.
Summing over all N molecules and dividing by the area L2 gives total pressure $P = (m / L^{3}) Σ v_{x}^{2}$.
Using vrms2 = vx2 + vy2 + vz2 = 3vx2, we arrive at $P = (1/3)nmv_{rms}^{2}$.
Result: $P = \frac{1}{3} \rho v_{rms}^{2}$
Mean Free Path of Gas Molecules
Conditions: Rigid spherical molecules of diameter d and number density n.
A molecule of diameter d sweeps out a cylindrical volume of cross-sectional area πd2 in moving with average speed v over time t.
The number of molecules swept through in time t is n(πd2v)t.
The average distance between successive collisions, known as the mean free path (λ), is the total distance traveled divided by the number of collisions.
Accounting for the relative velocities of colliding molecules introduces a correction factor of 1 / &sqrt;2.
Result: $\lambda = \frac{1}{\sqrt{2} \pi n d^{2}}$
Symbols denote standard thermodynamic parameters m/s
Speed corresponding to maximum probability
Average Kinetic Energy per Molecule
$$E = \frac{3}{2}k_{B}T$$
E = kinetic energy per molecule, kB = Boltzmann constant, T = temperature J
Valid for any ideal gas regardless of atomicity
Total Internal Energy of Ideal Gas
$$U = \frac{f}{2} nRT$$
f = degrees of freedom, n = number of moles, R = gas constant, T = temperature J
Equipartition theorem applied
Molar Heat Capacity at Constant Volume
$$C_{v} = \frac{f}{2} R$$
Cv = molar heat capacity, f = degrees of freedom, R = universal gas constant J/(mol K)
Rigid molecules
Mean Free Path
$$\lambda = \frac{1}{\sqrt{2}\pi n d^{2}}$$
$\lambda = mean free path$, n = number density, d = molecular diameter m
Hard sphere approximation
Graphs and experiments
Maxwell-Boltzmann Speed Distribution Curve
Draw: A skewed bell-shaped curve of fraction of molecules versus speed v, showing peaks for different temperatures T1 and T2 (T2 > T1).
Interpret: As temperature increases, the peak shifts to the right (higher most probable speed) and broadens, showing a wider range of molecular speeds.
Exam use: Directly tested in matching curves or identifying relative magnitudes of vmp, vavg, and vrms.
Pressure vs Temperature (Constant Volume)
Draw: A straight line passing through the origin when extrapolated backward, intersecting the temperature axis at absolute zero (-273.15 °C).
Interpret: Demonstrates Gay-Lussac's law and confirms that pressure is directly proportional to absolute temperature, vanishing at absolute zero.
Exam use: Used to test the definition of absolute zero temperature.
PV/RT vs P Isotherms for Real vs Ideal Gases
Draw: A horizontal straight line for ideal gas (Z=1) and curves showing significant deviations at high pressures for real gases.
Interpret: Illustrates the compressibility factor Z as a function of pressure, highlighting intermolecular forces and volume corrections.
Exam use: PGT questions on compressibility factor behavior at various pressure and temperature regimes.
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 parameters: T = 300 K, $M = 2 x 10^{-3} kg/mol$, $R = 8.314 J/(mol K)$.
Apply formula: vrms = sqrt(3RT / M).
Substitute values: vrms = sqrt((3 x 8.314 x 300) / (2 x 10-3)).
Calculate intermediate value: 3 x 8.314 x 300 = 7482.6; 7482.6 / 0.002 = 3,741,300.
Take square root: sqrt(3,741,300) = 1934.2 m/s.
Answer: 1934.2 m/s
PGT extension
At what temperature is the root-mean-square speed of oxygen molecules equal to 1000 m/s? Molar mass of oxygen is 32 g/mol.
Identify given parameters: vrms = 1000 m/s, $M = 32 x 10^{-3} kg/mol$, $R = 8.314 J/(mol K)$.
Rearrange rms speed formula to solve for temperature: $T = (v_{rms}^{2} M) / (3R)$.
Substitute values: $T = ((1000)^{2} x 32 x 10^{-3}) / (3 x 8.314)$.
Numerator calculation: 106 x 32 x 10-3 = 32,000.
Denominator calculation: 3 x 8.314 = 24.942.
Final division: 32,000 / 24.942 = 1282.98 K.
Answer: 1283 K
TGT core
Find the total internal energy of 2 moles of an ideal monoatomic gas at 27 °C. ($R = 8.31 J/mol K)$
Identify parameters: n = 2 moles, T = 27 °C = 300 K, monoatomic gas implies f = 3.
Use internal energy formula: $U = (f / 2) nRT$.
Substitute values: $U = (3 / 2) x 2 x 8.31 x 300$.
Simplify expression: U = 3 x 8.31 x 300 = 7479 J.
Answer: 7479 J
Assumptions and limitations
Assumption: Intermolecular forces are completely neglected except during elastic collisions. Limitation: Fails at high pressures and low temperatures where real gas intermolecular attractive forces become significant.
Assumption: Volume of molecules is point-like compared to container volume. Limitation: Fails at high densities where the excluded volume correction cannot be ignored.
Common misconceptions
Common error: Temperature is a measure of the total kinetic energy of all molecules in a container.
Correct: Temperature is a measure of the average translational kinetic energy per molecule.
Total kinetic energy depends on the total number of moles or molecules, whereas temperature is an intensive property reflecting mean kinetic energy per degree of freedom.
Common error: RMS speed and average speed of gas molecules are identical.
Correct: vrms is always greater than vavg.
Root-mean-square speed weighs higher speeds more heavily since it is based on the square root of mean squared speeds.
Quick revision
Pressure $P = (1/3) \rho v_{rms}^{2} establishes the link between macroscopic pressure$ and microscopic motion.
Degrees of freedom determine the specific heat ratio gamma via Mayer's relation and equipartition theorem.
The pressure of an ideal gas is given by $P = \frac{1}{3}nmv_{rms}^2 = \frac{1}{3}\rho v_{rms}^2$.
The root-mean-square speed of gas molecules is $v_{rms} = \sqrt{\frac{3RT}{M}}$, average speed is $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$, and most probable speed is $v_{mp} = \sqrt{\frac{2RT}{M}}$.
The order of characteristic speeds is always $v_{mp} < v_{avg} < v_{rms}$.
According to the Law of Equipartition of Energy, each active degree of freedom contributes $\frac{1}{2}k_B T$ per molecule.
Monoatomic, diatomic, and rigid polyatomic gases possess 3, 5, and 6 degrees of freedom respectively at room temperature.
Mean free path of gas molecules is given by $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$.
Glossary
Root Mean Square Speed ($v_{rms}$)The square root of the arithmetic mean of the squares of the speeds of the molecules in a gas.
Degrees of Freedom ($f$)The total number of independent coordinates required to specify the dynamical state of a mechanical system.
Mean Free Path ($\lambda$)The average distance traveled by a gas molecule between successive collisions.
Boltzmann Constant ($k_B$)The gas constant per molecule, defined as $R/N_A$.
Characteristic Speeds of Gas Molecules
Parameter
Most Probable Speed ($v_{mp}$)
Average Speed ($v_{avg}$)
Root Mean Square Speed ($v_{rms}$)
Formula
$\sqrt{\frac{2RT}{M}}$
$\sqrt{\frac{8RT}{\pi M}}$
$\sqrt{\frac{3RT}{M}}$
Numerical Factor
$\sqrt{2} \approx 1.414$
$\sqrt{\frac{8}{\pi}} \approx 1.596$
$\sqrt{3} \approx 1.732$
Relative Magnitude
Smallest
Intermediate
Largest
Practice Quiz
Choose an option to reveal the explanation.
1. What is the expression for the pressure of an ideal gas based on the kinetic theory?
Explanation: From the kinetic theory of gases, the pressure exerted by an ideal gas is given by $P = \frac{1}{3}nmv_{rms}^2$, where $n$ is numerical density, $m$ is mass per molecule, and $v_{rms}$ is the root mean square velocity.
2. Which of the following relations correctly represents the hierarchy of molecular speeds in a gas?
Explanation: The characteristic speeds are related as $v_{mp} = \sqrt{\frac{2RT}{M}}$ ($1.414$), $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$ ($1.596$), and $v_{rms} = \sqrt{\frac{3RT}{M}}$ ($1.732$), showing that $v_{rms} > v_{avg} > v_{mp}$.
3. What is the average kinetic energy per molecule of an ideal gas at absolute temperature $T$?
Explanation: The average translational kinetic energy per molecule of any ideal gas depends only on its temperature and is given by $E = \frac{3}{2}k_B T$.
4. How many degrees of freedom does a rigid diatomic gas molecule possess at room temperature?
Explanation: A rigid diatomic molecule has 3 translational and 2 rotational degrees of freedom, making a total of 5 degrees of freedom at room temperature.
5. According to the Law of Equipartition of Energy, what is the energy associated with each active degree of freedom per molecule?
Explanation: The Law of Equipartition of Energy states that each active degree of freedom contributes an energy of $\frac{1}{2}k_B T$ per molecule in thermal equilibrium.
6. What happens to the root-mean-square speed of gas molecules if the absolute temperature of the gas is quadrupled?
Explanation: Since $v_{rms} \propto \sqrt{T}$, increasing the absolute temperature $T$ by a factor of 4 increases $v_{rms}$ by a factor of $\sqrt{4} = 2$.
7. What is the expression for the mean free path ($\lambda$) of gas molecules?
Explanation: The mean free path $\lambda$ is given by $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$, where $n$ is number density and $d$ is the molecular diameter.
8. Which of the following parameters does the pressure of an ideal gas depend on?
Explanation: Gas pressure depends on mass density and mean square speed, and via the ideal gas law, on volume and temperature, but is completely independent of the shape and material of the container.
9. What is the molar heat capacity at constant volume ($C_v$) for a monoatomic ideal gas?
Explanation: For a monoatomic gas, $f = 3$. Using $C_v = \frac{f}{2}R$, we get $C_v = \frac{3}{2}R$.
10. If the molar mass of a gas is doubled at constant temperature, what happens to its $v_{rms}$?
Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, $v_{rms}$ is inversely proportional to the square root of the molar mass $M$. Doubling $M$ reduces $v_{rms}$ by a factor of $\sqrt{2}$.
11. Which assumption is NOT a postulate of the kinetic theory of ideal gases?
Explanation: An ideal gas assumes that there are no intermolecular forces of attraction or repulsion except during collisions.
12. What is the total internal energy of $n$ moles of a polyatomic gas with $f$ degrees of freedom at temperature $T$?
Explanation: By the equipartition theorem, the total internal energy for $n$ moles is given by $U = \frac{f}{2}nRT$.
13. How does the mean free path of a gas molecule vary with the numerical density ($n$) of the gas?
Explanation: From the formula $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$, the mean free path is inversely proportional to the numerical density $n$.
14. Which of the following characteristic speeds corresponds to the peak of the Maxwell-Boltzmann distribution curve?
Explanation: The most probable speed ($v_{mp}$) corresponds to the maximum of the Maxwell-Boltzmann distribution function where the fraction of molecules having that speed is highest.
15. Temperature is fundamentally a measure of which of the following?
Explanation: Temperature is an intensive property that directly measures the average translational kinetic energy per molecule of a system.
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 oxygen molecules at a certain temperature is $v$. If the temperature is doubled and the oxygen dissociates into atomic oxygen, what will be the new root-mean-square speed?
Explanation: For molecular oxygen ($O_2$), molar mass is $M$. When it dissociates into atomic oxygen ($O$), the molar mass becomes $M/2$. The rms speed is $v_{rms} = \sqrt{\frac{3RT}{M}}$. For the new state, $T' = 2T$ and $M' = M/2$. Thus, $v'_{rms} = \sqrt{\frac{3R(2T)}{M/2}} = \sqrt{4 \cdot \frac{3RT}{M}} = 2v$.
2. If the degrees of freedom of a gas molecule are $f$, the ratio of specific heats $\gamma = \frac{C_p}{C_v}$ is given by:
Explanation: We know $C_v = \frac{f}{2}R$ and $C_p = C_v + R = \left(\frac{f}{2} + 1\right)R$. Therefore, $\gamma = \frac{C_p}{C_v} = \frac{\frac{f}{2} + 1}{\frac{f}{2}} = 1 + \frac{2}{f}$.
3. At what temperature is the root-mean-square speed of hydrogen molecules equal to that of oxygen molecules at $300\text{ K}$?
4. The mean free path of a gas molecule depends on the molecular diameter $d$ as:
Explanation: From the formula $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$, the mean free path is inversely proportional to the square of the molecular diameter ($d^2$).
5. A gas mixture consists of 2 moles of oxygen and 4 moles of argon at temperature $T$. Assuming ideal behavior, what is the total internal energy of the system?
Explanation: Oxygen ($O_2$) is diatomic ($f_1 = 5$, $U_1 = \frac{5}{2}n_1RT = \frac{5}{2}(2)RT = 5RT$). Argon ($Ar$) is monoatomic ($f_2 = 3$, $U_2 = \frac{3}{2}n_2RT = \frac{3}{2}(4)RT = 6RT$). Total internal energy $U = 5RT + 6RT = 11RT$.
6. The ratio of the most probable speed to the root-mean-square speed for an ideal gas is:
Explanation: Most probable speed is $v_{mp} = \sqrt{\frac{2RT}{M}}$ and rms speed is $v_{rms} = \sqrt{\frac{3RT}{M}}$. Their ratio is $\frac{v_{mp}}{v_{rms}} = \sqrt{\frac{2}{3}}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the root-mean-square velocity of hydrogen gas molecules at $300\text{ K}$ is $1900\text{ m/s}$, what is the rms velocity of helium gas molecules at the same temperature?
Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, for the same temperature $T$, $v_{rms} \propto \frac{1}{\sqrt{M}}$. Molar mass of $H_2$ is $2\text{ g/mol}$ and of $He$ is $4\text{ g/mol}$. Thus, $v_{He} = v_{H_2} \sqrt{\frac{M_{H_2}}{M_{He}}} = 1900 \sqrt{\frac{2}{4}} = \frac{1900}{\sqrt{2}}\text{ m/s}$.
2. What is the degrees of freedom for a non-linear triatomic gas molecule at room temperature?
Explanation: A non-linear triatomic molecule has 3 translational and 3 rotational degrees of freedom, giving a total of 6 degrees of freedom at room temperature.
3. The kinetic energy of translation of $20\text{ g}$ of oxygen at $0^\circ\text{C}$ is approximately: (Given $R = 8.31\text{ J/(mol K)}$)
4. Which of the following statements about the Maxwell-Boltzmann speed distribution curve is correct?
Explanation: As temperature increases, molecules move faster, causing the peak of the Maxwell-Boltzmann curve to shift to the right and flatten out while keeping the total area under the curve constant.
5. If pressure of a gas is doubled keeping the volume constant, the root-mean-square speed of the molecules:
Explanation: Since $P = \frac{1}{3}\rho v_{rms}^2 = \frac{1}{3}\frac{M}{V}v_{rms}^2$, if $V$ is constant and $P$ is doubled, $v_{rms}^2$ must double, meaning $v_{rms}$ increases by a factor of $\sqrt{2}$.
6. What is the ratio of root-mean-square speed to average speed for gas molecules?
Explanation: $v_{rms} = \sqrt{\frac{3RT}{M}}$ and $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$. The ratio $\frac{v_{rms}}{v_{avg}} = \sqrt{\frac{3}{\frac{8}{\pi}}} = \sqrt{\frac{3\pi}{8}}$.
7. At what temperature will the rms speed of oxygen molecules be twice its value at $27^\circ\text{C}$?
Explanation: Initial temperature $T_1 = 27 + 273 = 300\text{ K}$. We need $v_{rms} \propto \sqrt{T}$ to double, which means temperature must quadruple ($T_2 = 4 \times 300 = 1200\text{ K}$).
8. A closed rigid container has an ideal gas. If heat is supplied to increase its temperature, what remains constant?
Explanation: Since the container is rigid and closed, its volume remains strictly constant during the heating process.
9. The mean free path of gas molecules is inversely proportional to:
Explanation: From $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$, the mean free path is inversely proportional to the cross-sectional area $\pi d^2$ and number density $n$.
10. According to the Law of Equipartition of Energy, the total internal energy of 1 mole of a monoatomic gas at absolute temperature $T$ is:
Explanation: For 1 mole of a monoatomic gas, $f = 3$. The internal energy is $U = \frac{f}{2}RT = \frac{3}{2}RT$.