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Thermal Physics

Root Mean Square Velocity of Gas Molecules

Root mean square (RMS) velocity is the square root of the mean of the squares of the velocities of individual gas molecules. It provides a direct measure of the average translational kinetic energy of gas molecules at a given temperature.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Calculation of RMS velocity using temperature, molecular mass, pressure, and density; comparative problems involving different gases and temperatures.

PGT extension

Derivation from kinetic theory postulates, relationship with Maxwell-Boltzmann distribution, degrees of freedom, and translational kinetic energy per mole.

Prerequisites

Postulates of kinetic theory of gases
Concept of absolute temperature
Ideal gas equation and molar mass

Definition and Physical Significance

RMS velocity (v_rms) is defined mathematically as v_rms = √(〈v²〉).
It accounts for the directional cancellation of random molecular velocities where the vector average velocity is zero.
It is always greater than the average velocity and the most probable velocity for a given Maxwellian gas.

Dependence on Temperature and Mass

v_rms is directly proportional to the square root of absolute temperature (√T).
v_rms is inversely proportional to the square root of the molar mass or molecular weight (√M).
It is independent of pressure and volume of the given mass of an ideal gas at constant temperature.

Relation with Kinetic Energy

The translational kinetic energy per mole of an ideal gas is expressed as $E = (3/2)RT$.
Equating macroscopic pressure equation to molecular parameters links v_rms directly to thermal energy.
At absolute zero (T = 0 K), the RMS velocity of gas molecules theoretically vanishes.

Comparison with Other Speeds

Most probable velocity v_mp : v_rms : average velocity v_av = √2 : √3 : √(8/π) ≈ 1.414 : 1.732 : 1.596.
The hierarchical order is always v_mp < v_av < v_rms.
All three characteristic speeds scale linearly with √(T/M).

Derivations

Derivation of RMS Velocity from Kinetic Theory Pressure Equation

Conditions: Ideal gas, N molecules in volume V, isotropic random motion

Start with the kinetic theory pressure expression: $P = (1/3)(NM/V)\langle v^{2}\rangle or P = (1/3)\rhov_{rms}^{2}$ where ρ is density.
Substitute P$V = nRT = (N/N_{A})RT = NkT$, where k is the Boltzmann constant.
Equate the two expressions: (1/3)(N/V)m ⟨v²$\rangle = (N/V)kT$, where m is the mass of a single molecule (M/N_A).
Solve for v_rms = √⟨v²$\rangle = \sqrt{3kT/m} = \sqrt{3RT/M}$.

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

Derivation from Law of Equipartition of Energy

Conditions: Monatomic ideal gas with 3 translational degrees of freedom

According to the law of equipartition of energy, each translational degree of freedom contributes (1/2)kT to the average energy of a molecule.
Total average kinetic energy of a single molecule with 3 degrees of freedom is ⟨E$ \rangle = 3(\frac{1}{2}kT) = (3/2)kT$.
Equate this to the microscopic kinetic energy formula: (1/2)m⟨v²$\rangle = (3/2)kT$.
Cancel mass and multiply by 2 to isolate ⟨v²$\rangle = 3kT/m$, then take the square root.

Result: $v_rms = \sqrt{\frac{3kT}{m}}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
RMS velocity in terms of Temperature and Molar Mass	$$v_{rms} = \sqrt{\frac{3RT}{M}}$$	R = universal gas constant, T = absolute temperature, M = molar mass m s^-1	Ideal gas
RMS velocity in terms of Boltzmann Constant and Molecular Mass	$$v_{rms} = \sqrt{\frac{3kT}{m}}$$	k = Boltzmann constant, T = absolute temperature, m = mass of single molecule m s^-1	Ideal gas
RMS velocity in terms of Pressure and Density	$$v_{rms} = \sqrt{\frac{3P}{\rho}}$$	P = pressure, $\rho = density of gas$ m s^-1	Constant temperature and mass
Most Probable Velocity	$$v_{mp} = \sqrt{\frac{2RT}{M}}$$	R = gas constant, T = temperature, M = molar mass m s^-1	Maxwell-Boltzmann distribution
Average Velocity	$$v_{av} = \sqrt{\frac{8RT}{\pi M}}$$	R = gas constant, T = temperature, M = molar mass m s^-1	Maxwell-Boltzmann distribution
Ratio of Speeds	$$v_{mp} : v_{av} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}$$	v_{mp}, v_{av}, v_{rms} are most probable, average, and rms velocities Dimensionless ratio	Same gas and temperature
Mean Square Velocity	$$\langle v^2 \rangle = \frac{3RT}{M}$$	⟨v²$\rangle = average of squared speeds$ m² s^-2	Equilibrium state
Temperature Dependence Ratio	$$\frac{v_{rms1}}{v_{rms2}} = \sqrt{\frac{T_1}{T_2}}$$	v_{rms1}, v_{rms2} at temperatures T_1, T_2 for same gas Dimensionless	Same chemical species

Graphs and experiments

Maxwell-Boltzmann Speed Distribution Curve

Draw: Probability distribution curve of molecular speeds showing asymmetric bell shape with peaks for v_mp, v_av, and v_rms.

Interpret: Visualizes the fraction of molecules possessing various speeds, highlighting that v_rms lies towards the higher-speed tail compared to the peak.

Exam use: Frequently tested in PGT to identify the relative positions of the three characteristic speeds on the distribution curve.

Effect of Temperature on Speed Distribution

Draw: Two Maxwellian curves on the same axes: one at lower temperature T₁ (taller, narrower peak) and one at higher temperature T₂ (flattened, broader, shifted right).

Interpret: Higher temperature broadens the distribution and shifts the RMS velocity to a higher value due to increased kinetic energy.

Exam use: Tested via conceptual multiple-choice questions regarding how peak height and RMS velocity change with heating.

Stern's Experiment / Molecular Beam Velocity Verification

Draw: Schematic of rotating coaxial cylinders with an oven emitting atomic beams of metal (like silver) deposited on a strip.

Interpret: Directly measured the velocity distribution of atoms in a thermal vapor beam, confirming theoretical molecular speeds.

Exam use: Contextual background for experimental validation of kinetic theory in advanced PGT segments.

Solved numericals

TGT core

Calculate the root mean square velocity of oxygen molecules (O₂) at 27 °C. Given $R = 8.31 J mol^{-1} K^{-1}$ and molar mass of oxyge$n = 32 \times 10^{-3} kg mol^{-1}$.

Identify given values: $T = 27 °C = 27 + 273 = 300 K$, $M = 32 \times 10^{-3} kg mol^{-1}$, $R = 8.31 J mol^{-1} K^{-1}$.
Substitute values into the formula v_rms = √(3RT/M).
v_rms = √(3 × 8.31 × 300 / (32 × 10^-3)) = √(7479 / 0.032) = √(233718.75).
Calculate the square root: v_rms ≈ 483.4 m s^-1.

Answer: 483.4 m s^-1

PGT extension

At what temperature will the root mean square velocity of hydrogen molecules be equal to that of oxygen molecules at 300 K? Molar mass of H₂ = 2 g mol^-1, O₂ = 32 g mol^-1.

Recall that v_rms = √(3RT/M), meaning v_rms ∝ √(T/M).
Set the RMS velocity of hydrogen equal to that of oxygen: √(3RT_H / M_H) = √(3RT_O / M_O).
Square both sides and simplify: T_H / M_H = T_O / M_O.
Substitute known values: T_H / $2 = 300 / 32$.
Solve for T_H: T_H = 2 × 300 / $32 = 600 / 32 = 18.75 K$.

Answer: 18.75 K

TGT core

If the absolute temperature of a fixed mass of an ideal gas is quadrupled, what will be the percentage increase in its root mean square velocity?

Initial velocity v₁ = √(3RT₁/M) and final velocity v₂ = √(3R(4T₁)/M) = 2 √(3RT₁/M) = 2v₁.
The new velocity is twice the original velocity.
Calculate percentage increase: ((v₂ - v₁) / v₁) × 100% = ((2v₁ - v₁) / v₁) × 100% = 100%.
A factor of 2 increase represents a 100% increase from the base value.

Answer: 100%

Assumptions and limitations

Assumes gases obey the ideal gas law (PV = nRT), which breaks down at high pressures and low temperatures due to intermolecular forces and finite molecular volume.
Molecules are treated as point masses ignoring rotational and vibrational kinetic energy components during pure translational RMS calculations.

Common misconceptions

Common error: RMS velocity depends on the pressure of the gas container.

Correct: RMS velocity is independent of pressure at constant temperature.

When pressure increases at constant temperature, density increases proportionally, keeping the ratio P/ρ and hence v_rms constant.

Common error: All molecules in a gas sample travel at the RMS velocity.

Correct: Molecules possess a continuous distribution of speeds ranging from zero to infinity.

v_rms is only a statistical root-mean-square aggregate value, not the speed of every individual molecule.

Common error: RMS velocity and average velocity are numerically identical.

Correct: v_rms is always greater than the average velocity.

Squaring speeds before averaging heavily weights higher-speed molecules, making v_rms larger than the direct arithmetic mean speed.

Quick revision

v_rms is proportional to √T and inversely proportional to √M.
RMS velocity in terms of Temperature and Molar Mass: $v_{rms} = \sqrt{\frac{3RT}{M}}$
RMS velocity in terms of Boltzmann Constant and Molecular Mass: $v_{rms} = \sqrt{\frac{3kT}{m}}$

Must remember

The root mean square velocity is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$ or $v_{rms} = \sqrt{\frac{3kT}{m}}$ or $v_{rms} = \sqrt{\frac{3P}{\rho}}$.
For a given mass of an ideal gas at constant temperature, $v_{rms}$ is independent of pressure and volume.
The hierarchical order of molecular speeds is always $v_{mp} < v_{av} < v_{rms}$ with ratio $\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}$.
At absolute zero ($T = 0$ K), the root mean square velocity of gas molecules theoretically vanishes.
RMS velocity accounts for the directional cancellation of random molecular velocities where the vector average velocity is zero.

Glossary

Root Mean Square Velocity ($v_{rms}$)The square root of the arithmetic mean of the squares of the speeds of the molecules in a gas sample.

Most Probable Velocity ($v_{mp}$)The speed possessed by the maximum number of molecules in a Maxwell-Boltzmann distribution.

Average Velocity ($v_{av}$)The arithmetic mean of the speeds of all molecules in a gas sample.

Boltzmann Constant ($k$)The gas constant per molecule, defined as $R/N_A$.

Comparison of Characteristic Molecular Speeds

Speed Type	Formula	Relative Magnitude
Most Probable Velocity ($v_{mp}$)	$\sqrt{\frac{2RT}{M}}$	$\approx 1.414$
Average Velocity ($v_{av}$)	$\sqrt{\frac{8RT}{\pi M}}$	$\approx 1.596$
RMS Velocity ($v_{rms}$)	$\sqrt{\frac{3RT}{M}}$	$\approx 1.732$

Practice Quiz

Choose an option to reveal the explanation.

1. What is the mathematical definition of root mean square velocity ($v_{rms}$) for gas molecules?

Explanation: RMS velocity is defined as the square root of the mean of the squared speeds, denoted as $\sqrt{\langle v^2 \rangle}$.

2. How does the root mean square velocity of an ideal gas depend on its absolute temperature ($T$)?

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

3. At constant temperature, if the pressure of a given mass of an ideal gas is doubled, what happens to its $v_{rms}$?

Explanation: RMS velocity is independent of pressure and volume at constant temperature because density increases proportionally with pressure, keeping $P/\rho$ constant.

4. What is the correct hierarchical order of most probable speed ($v_{mp}$), average speed ($v_{av}$), and RMS speed ($v_{rms}$)?

Explanation: The speeds follow the ratio $\sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$, which means $v_{mp}$ ($1.414$) < $v_{av}$ ($1.596$) < $v_{rms}$ ($1.732$).

5. What is the ratio of $v_{mp} : v_{av} : v_{rms}$?

Explanation: The exact ratio derived from Maxwell-Boltzmann distribution is $\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}$.

6. What happens to the RMS velocity of gas molecules at absolute zero ($T = 0$ K)?

Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, putting $T = 0$ K makes the RMS velocity theoretically zero.

7. If the molar mass of a gas is quadrupled at a constant temperature, its RMS velocity will:

Explanation: Since $v_{rms}$ is inversely proportional to $\sqrt{M}$, increasing $M$ by a factor of $4$ reduces $v_{rms}$ by a factor of $\sqrt{4} = 2$ (halved).

8. Which formula expresses RMS velocity in terms of pressure ($P$) and density ($\rho$)?

Explanation: Using $P = \frac{1}{3} \rho v_{rms}^2$, we get $v_{rms} = \sqrt{\frac{3P}{\rho}}$.

9. Why is the vector average velocity of gas molecules zero in a stationary container, while $v_{rms}$ is not?

Explanation: Vector average accounts for direction, summing to zero due to random isotropy. RMS velocity squares the speeds, making all terms positive.

10. If the temperature of an ideal gas is increased from $27^{\circ}\text{C}$ to $327^{\circ}\text{C}$, the ratio of their RMS velocities $v_{rms1} / v_{rms2}$ is:

Explanation: Temperatures in Kelvin are $T_1 = 300$ K and $T_2 = 600$ K. Ratio $\frac{v_{rms1}}{v_{rms2}} = \sqrt{\frac{300}{600}} = \frac{1}{\sqrt{2}}$.

11. How is the RMS velocity of gas molecules related to the Boltzmann constant ($k$) and the mass of a single molecule ($m$)?

Explanation: Substituting $R = N_A k$ and $M = N_A m$ into $v_{rms} = \sqrt{\frac{3RT}{M}}$ yields $v_{rms} = \sqrt{\frac{3kT}{m}}$.

12. Which of the following statements is a misconception regarding RMS velocity?

Explanation: RMS velocity is independent of pressure at constant temperature because changes in pressure are offset by proportional changes in density.

13. What is the dimension of RMS velocity?

Explanation: RMS velocity is a speed, so its dimensional formula is $[L T^{-1}]$ (meters per second).

14. Two different gases, Hydrogen ($M = 2$) and Oxygen ($M = 32$), are at the same temperature. What is the ratio of their RMS velocities ($v_{H2} / v_{O2}$)?

Explanation: $\frac{v_{H2}}{v_{O2}} = \sqrt{\frac{M_{O2}}{M_{H2}}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4:1$.

15. What does the mean square velocity $\langle v^2 \rangle$ represent physically?

Explanation: Mean square velocity $\langle v^2 \rangle$ is defined mathematically as the average of the squared speeds of the gas molecules.

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 RMS velocity 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 temperature $T_1$ and $T_2 = 300$ K: $\sqrt{\frac{T_1}{2}} = \sqrt{\frac{300}{32}} \implies T_1 = 300 \times \frac{2}{32} = \frac{300}{16} = 18.75$ K.

2. The root mean square speed of oxygen molecules at $0^{\circ}\text{C}$ is approximately $460$ m/s. What will be the RMS speed of oxygen molecules at $273^{\circ}\text{C}$?

Explanation: Temperatures are $T_1 = 273$ K and $T_2 = 546$ K. $v_{rms2} = v_{rms1} \sqrt{\frac{546}{273}} = 460 \times \sqrt{2}$ m/s.

3. If the root mean square velocity of gas molecules is $v$, and its absolute temperature is doubled while the molar mass is halved, the new RMS velocity becomes:

Explanation: New velocity $v' = \sqrt{\frac{3R(2T)}{(M/2)}} = \sqrt{4 \cdot \frac{3RT}{M}} = 2 \sqrt{\frac{3RT}{M}} = 2v$.

4. The translational kinetic energy of $1$ mole of an ideal gas is given by $E = \frac{3}{2}RT$. Using this, the RMS velocity can be expressed as:

Explanation: Since $E = \frac{3}{2}RT$, we have $RT = \frac{2E}{3}$. Substituting this into $v_{rms} = \sqrt{\frac{3RT}{M}}$ gives $v_{rms} = \sqrt{\frac{3(2E/3)}{M}} = \sqrt{\frac{2E}{M}}$ ... wait, let's recheck: $3RT = 2E$, so $v_{rms} = \sqrt{\frac{2E}{M}}$. Let's verify option alignment. Ah, $3RT = 2E$, so $v_{rms} = \sqrt{\frac{2E}{M}}$ which corresponds to option A if correctly indexed. Let's correct correct_index to 0.

5. The density of a gas at $1.01 \times 10^5$ N/m$^2$ pressure and $0^{\circ}\text{C}$ is $1.29$ kg/m$^3$. What is its RMS velocity?

Explanation: $v_{rms} = \sqrt{\frac{3P}{\rho}} = \sqrt{\frac{3 \times 1.01 \times 10^5}{1.29}} = \sqrt{2.348 \times 10^5} \approx 484.5$ m/s.

6. What is the ratio of the RMS speed of helium atoms ($M = 4$ g/mol) to that of hydrogen molecules ($M = 2$ g/mol) at the same temperature?

Explanation: $\frac{v_{He}}{v_{H2}} = \sqrt{\frac{M_{H2}}{M_{He}}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}}$.

Timed Topic Test

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

1. Which of the following molecular speeds is directly used in calculating the total translational kinetic energy of an ideal gas sample?

Explanation: Kinetic energy per mole is $E = \frac{1}{2} M \langle v^2 \rangle$, which directly links energy to $v_{rms}$.

2. If the absolute temperature of a gas is increased four times, the root mean square velocity of the gas molecules will:

Explanation: Since $v_{rms} \propto \sqrt{T}$, increasing $T$ by $4$ times increases $v_{rms}$ by $\sqrt{4} = 2$ times.

3. Which gas molecules will have the highest RMS velocity at room temperature?

Explanation: RMS velocity is inversely proportional to the square root of molar mass ($\sqrt{M}$). Hydrogen has the lowest molar mass ($2$ g/mol), hence the highest $v_{rms}$.

4. In a Maxwellian distribution of speeds, what fraction of molecules travel at speeds greater than infinity?

Explanation: The probability density function approaches zero asymptotically as speed approaches infinity, so the fraction is zero.

5. Which expression correctly relates the Boltzmann constant ($k$), gas constant ($R$), and Avogadro number ($N_A$)?

Explanation: The universal gas constant $R$ is equal to the product of the Boltzmann constant $k$ and Avogadro's number $N_A$ ($R = k N_A$).

6. Under what condition is the root mean square velocity of a real gas expected to deviate significantly from ideal gas behavior?

Explanation: Low temperature and high pressure cause real gases to deviate from ideal behavior due to intermolecular forces and finite molecular volumes.

7. If $v_{rms}$ of a gas molecule is $500$ m/s at $27^{\circ}\text{C}$, what would be its approximate RMS velocity at $327^{\circ}\text{C}$?

Explanation: $T_1 = 300$ K, $T_2 = 600$ K. $v_{rms2} = 500 \times \sqrt{\frac{600}{300}} = 500 \times \sqrt{2} \approx 500 \times 1.414 = 707$ m/s.

8. Why is the root mean square velocity preferred over average velocity in kinetic theory derivations?

Explanation: Pressure derivation involves the average of squared velocities ($\langle v^2 \rangle$) from particle momentum changes upon wall collisions, making $v_{rms}$ directly relevant.

9. What is the ratio of average velocity ($v_{av}$) to RMS velocity ($v_{rms}$) for an ideal gas?

Explanation: $v_{av} = \sqrt{\frac{8RT}{\pi M}}$ and $v_{rms} = \sqrt{\frac{3RT}{M}}$. Their ratio $\frac{v_{av}}{v_{rms}} = \sqrt{\frac{8}{3\pi}}$.

10. Which of the following parameters does NOT affect the RMS velocity of gas molecules in a closed rigid container?

Explanation: RMS velocity is independent of the volume and pressure of the container; it depends solely on temperature and molecular mass.

Root Mean Square Velocity of Gas Molecules

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Definition and Physical Significance

Dependence on Temperature and Mass

Relation with Kinetic Energy

Comparison with Other Speeds

Derivations

Derivation of RMS Velocity from Kinetic Theory Pressure Equation

Derivation from Law of Equipartition of Energy

Formula sheet

Graphs and experiments

Maxwell-Boltzmann Speed Distribution Curve

Effect of Temperature on Speed Distribution

Stern's Experiment / Molecular Beam Velocity Verification

Solved numericals

Calculate the root mean square velocity of oxygen molecules (O2) at 27 °C. Given $R = 8.31 J mol^{-1} K^{-1}$ and molar mass of oxyge$n = 32 \times 10^{-3} kg mol^{-1}$.

At what temperature will the root mean square velocity of hydrogen molecules be equal to that of oxygen molecules at 300 K? Molar mass of H2 = 2 g mol-1, O2 = 32 g mol-1.

If the absolute temperature of a fixed mass of an ideal gas is quadrupled, what will be the percentage increase in its root mean square velocity?

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Comparison of Characteristic Molecular Speeds

Practice Quiz

1. What is the mathematical definition of root mean square velocity ($v_{rms}$) for gas molecules?

2. How does the root mean square velocity of an ideal gas depend on its absolute temperature ($T$)?

3. At constant temperature, if the pressure of a given mass of an ideal gas is doubled, what happens to its $v_{rms}$?

4. What is the correct hierarchical order of most probable speed ($v_{mp}$), average speed ($v_{av}$), and RMS speed ($v_{rms}$)?

5. What is the ratio of $v_{mp} : v_{av} : v_{rms}$?

6. What happens to the RMS velocity of gas molecules at absolute zero ($T = 0$ K)?

7. If the molar mass of a gas is quadrupled at a constant temperature, its RMS velocity will:

8. Which formula expresses RMS velocity in terms of pressure ($P$) and density ($\rho$)?

9. Why is the vector average velocity of gas molecules zero in a stationary container, while $v_{rms}$ is not?

10. If the temperature of an ideal gas is increased from $27^{\circ}\text{C}$ to $327^{\circ}\text{C}$, the ratio of their RMS velocities $v_{rms1} / v_{rms2}$ is:

11. How is the RMS velocity of gas molecules related to the Boltzmann constant ($k$) and the mass of a single molecule ($m$)?

12. Which of the following statements is a misconception regarding RMS velocity?

13. What is the dimension of RMS velocity?

14. Two different gases, Hydrogen ($M = 2$) and Oxygen ($M = 32$), are at the same temperature. What is the ratio of their RMS velocities ($v_{H2} / v_{O2}$)?

15. What does the mean square velocity $\langle v^2 \rangle$ represent physically?

PYQ Patterns

1. At what temperature will the RMS velocity of hydrogen molecules be equal to that of oxygen molecules at $300$ K?

2. The root mean square speed of oxygen molecules at $0^{\circ}\text{C}$ is approximately $460$ m/s. What will be the RMS speed of oxygen molecules at $273^{\circ}\text{C}$?

3. If the root mean square velocity of gas molecules is $v$, and its absolute temperature is doubled while the molar mass is halved, the new RMS velocity becomes:

4. The translational kinetic energy of $1$ mole of an ideal gas is given by $E = \frac{3}{2}RT$. Using this, the RMS velocity can be expressed as:

5. The density of a gas at $1.01 \times 10^5$ N/m$^2$ pressure and $0^{\circ}\text{C}$ is $1.29$ kg/m$^3$. What is its RMS velocity?

6. What is the ratio of the RMS speed of helium atoms ($M = 4$ g/mol) to that of hydrogen molecules ($M = 2$ g/mol) at the same temperature?

Timed Topic Test

1. Which of the following molecular speeds is directly used in calculating the total translational kinetic energy of an ideal gas sample?

2. If the absolute temperature of a gas is increased four times, the root mean square velocity of the gas molecules will:

3. Which gas molecules will have the highest RMS velocity at room temperature?

4. In a Maxwellian distribution of speeds, what fraction of molecules travel at speeds greater than infinity?

5. Which expression correctly relates the Boltzmann constant ($k$), gas constant ($R$), and Avogadro number ($N_A$)?

6. Under what condition is the root mean square velocity of a real gas expected to deviate significantly from ideal gas behavior?

7. If $v_{rms}$ of a gas molecule is $500$ m/s at $27^{\circ}\text{C}$, what would be its approximate RMS velocity at $327^{\circ}\text{C}$?

8. Why is the root mean square velocity preferred over average velocity in kinetic theory derivations?

9. What is the ratio of average velocity ($v_{av}$) to RMS velocity ($v_{rms}$) for an ideal gas?

10. Which of the following parameters does NOT affect the RMS velocity of gas molecules in a closed rigid container?

Calculate the root mean square velocity of oxygen molecules (O₂) at 27 °C. Given $R = 8.31 J mol^{-1} K^{-1}$ and molar mass of oxyge$n = 32 \times 10^{-3} kg mol^{-1}$.

At what temperature will the root mean square velocity of hydrogen molecules be equal to that of oxygen molecules at 300 K? Molar mass of H₂ = 2 g mol^-1, O₂ = 32 g mol^-1.