Grade 11 : Physics - Kinetic Theory (Set 4)

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Grade 11 : Physics - Kinetic Theory (Set 4)— Questions & Detailed Solutions

A sample of Nitrogen gas ($N_2$, molar mass $M = 28\text{ g/mol}$) is initially at an absolute temperature of 300 K with a root-mean-square speed $v_{rms} = 500m/s$. The sample is heated at constant volume to 1200 K.
What is the new root-mean-square speed ($v_{rms}'$) of the Nitrogen gas molecules at 1200 K?

(A)

500 m/s

(B)

707 m/s

(C)

1000 m/s

(D)

2000 m/s

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:The rms speed is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$, which implies $v_{rms} \propto \sqrt{T}$.
Ratio: $\frac{v_{rms}'}{v_{rms}} = \sqrt{\frac{1200}{300}} = \sqrt{4} = 2$.
Therefore, $v_{rms}' = 2 \times 500m/s = 1000m/s$.

Container geometry: A closed cubic vessel filled with an ideal gas of density $\rho = 1.2\text{ kg/m}^3$. The rms velocity of the gas molecules is measured as $v_{rms} = 500m/s$.
What is the pressure exerted by the gas molecules on the container walls?

(A)

$0.5 \times 10^5\text{ Pa}$

(B)

$1.0 \times 10^5\text{ Pa}$

(C)

$1.5 \times 10^5\text{ Pa}$

(D)

$3.0 \times 10^5\text{ Pa}$

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation:From kinetic theory, pressure is $P = \frac{1}{3}\rho v_{rms}^2$.
Substituting values: $P = \frac{1}{3} \times 1.2 \times (500)^2 = 0.4 \times 250000 = 100,000\text{ Pa} = 1.0 \times 10^5\text{ Pa}$.

A rigid cylinder contains 2 moles of Hydrogen gas ($H_2$) at temperature $T$. At this moderate temperature, vibrational modes are inactive, so each molecule possesses 5 active degrees of freedom.
What is the total internal energy ($U$) of the gas sample?

(A)

$3 RT$

(B)

$5 RT$

(C)

$6 RT$

(D)

$7 RT$

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation:The total internal energy is $U = \frac{f}{2} n R T$, where $f = 5$ degrees of freedom and $n = 2$ moles.
$U = \frac{5}{2} \times 2 \times R T = 5 RT$.

The mean free path of gas molecules in a chamber is $\lambda$. Due to chemical modification, the effective molecular diameter $d$ is doubled while keeping the number density $n$ of the molecules constant.
What is the new mean free path $\lambda'$ of the gas?

(A)

Doubled

(B)

Halved

(C)

Reduced to one-fourth

(D)

Quadrupled

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:The mean free path is $\lambda = \frac{1}{\sqrt{2} \pi n d^2}$, so $\lambda \propto \frac{1}{d^2}$.
If $d' = 2d$, then $\lambda' = \frac{\lambda}{2^2} = \frac{\lambda}{4}$.

Gas Sample Gas Species Temperature ($K$) Molar Mass ($g/mol$)
Sample A Helium (He) 300 4
Sample B Neon (Ne) 300 20
Sample C Argon (Ar) 300 40
Based on the table above, which gas sample has the highest average translational kinetic energy per molecule?

Gas Sample	Gas Species	Temperature ($K$)	Molar Mass ($g/mol$)
Sample A	Helium (He)	300	4
Sample B	Neon (Ne)	300	20
Sample C	Argon (Ar)	300	40

(A)

Sample A

(B)

Sample B

(C)

Sample C

(D)

All three samples have equal average translational kinetic energy

View Correct Answer & Detailed Solution

✓ Correct Option: (D)

Explanation:The average translational kinetic energy per molecule is $E_k = \frac{3}{2} k_B T$, which depends strictly on absolute temperature $T$ and is independent of mass.
Since all three gases are at 300 K, their average translational kinetic energies per molecule are identical.

Maxwell-Boltzmann velocity distribution curve: A plot showing three characteristic speeds on the horizontal axis ordered left-to-right as $v_1 < v_2 < v_3$.
Which set correctly identifies $v_1, v_2,$ and $v_3$?

(A)

$v_1 = v_{mp},\; v_2 = v_{avg},\; v_3 = v_{rms}$

(B)

$v_1 = v_{rms},\; v_2 = v_{avg},\; v_3 = v_{mp}$

(C)

$v_1 = v_{avg},\; v_2 = v_{mp},\; v_3 = v_{rms}$

(D)

$v_1 = v_{mp},\; v_2 = v_{rms},\; v_3 = v_{avg}$

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation:The three speeds are $v_{mp} = \sqrt{2RT/M} \approx 1.414 \sqrt{RT/M}$, $v_{avg} = \sqrt{8RT/(\pi M)} \approx 1.596 \sqrt{RT/M}$, and $v_{rms} = \sqrt{3RT/M} \approx 1.732 \sqrt{RT/M}$.
Thus $v_{mp} < v_{avg} < v_{rms}$, which corresponds to $v_1, v_2, v_3$.

What is the theoretical ratio of most probable speed ($v_{mp}$), average speed ($v_{avg}$), and root-mean-square speed ($v_{rms}$) for an ideal gas?

(A)

$1 : 1.128 : 1.225$

(B)

$1.225 : 1.128 : 1$

(C)

$1 : 1.225 : 1.128$

(D)

$1.128 : 1 : 1.225$

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation:The ratio $v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} = 1 : \sqrt{\frac{4}{\pi}} : \sqrt{\frac{3}{2}} \approx 1 : 1.128 : 1.225$.

A thermal vessel contains a gas mixture of 1 mole of Helium (monoatomic, $f=3$) and 3 moles of Oxygen (diatomic rigid, $f=5$) in thermal equilibrium at temperature $T$.
What is the total internal energy of this gas mixture?

(A)

$6 RT$

(B)

$7.5 RT$

(C)

$9 RT$

(D)

$10.5 RT$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:$U_{total} = U_{He} + U_{O2} = n_1 \frac{f_1}{2} RT + n_2 \frac{f_2}{2} RT$.
$U_{total} = 1 \times \left(\frac{3}{2}\right) RT + 3 \times \left(\frac{5}{2}\right) RT = 1.5 RT + 7.5 RT = 9 RT$.

An ideal gas is sealed in a container of fixed volume $V$. The absolute temperature of the gas is increased from $T$ to $3T$.
By what factor does the rms speed of the gas molecules increase?

(A)

Triples

(B)

Increases by a factor of $\sqrt{3}$

(C)

Increases by a factor of 1.5

(D)

Remains unchanged

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation:$v_{rms} = \sqrt{\frac{3RT}{M}}$, which means $v_{rms} \propto \sqrt{T}$.
When absolute temperature is tripled ($T \to 3T$), $v_{rms}$ increases by $\sqrt{3}$.

Q10

Statement I: The mean free path of gas molecules is inversely proportional to the mass density of the gas.
Statement II: At constant temperature, increasing the pressure of an ideal gas increases its mean free path.
Which of the following statements is correct?

(A)

Both Statement I and Statement II are true

(B)

Both Statement I and Statement II are false

(C)

Statement I is true, but Statement II is false

(D)

Statement I is false, but Statement II is true

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:Mean free path $\lambda = \frac{m}{\sqrt{2} \pi \rho d^2}$, so $\lambda \propto 1/\rho$ (Statement I is true).
At constant temperature, $P = n k_B T$, so increasing $P$ increases number density $n$, which decreases $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$ (Statement II is false).

Q11

What is the total translational kinetic energy of 1 mole of a monoatomic ideal gas at $T = 300\text{ K}$? (Take $R = 8.314\text{ J/(mol}\cdot\text{K)}$)

(A)

1.25 kJ

(B)

2.49 kJ

(C)

3.74 kJ

(D)

4.98 kJ

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:Molar translational kinetic energy $E_k = \frac{3}{2} R T$.
$E_k = 1.5 \times 8.314 \times 300 = 3741.3\text{ J} \approx 3.74\text{ kJ}$.

Q12

In the real gas van der Waals equation $\left(P + \frac{a}{V^2}\right)(V - b) = RT$, what physical correction does constant $a$ account for?

(A)

The finite volume occupied by gas molecules

(B)

The intermolecular attractive forces

(C)

The average translational velocity

(D)

The collision frequency with container walls

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation:The constant $a$ corrects pressure for intermolecular forces of attraction, whereas $b$ corrects for the finite volume occupied by the gas molecules.

Q13

An air bubble of initial volume $V_0$ forms at the bottom of a lake where absolute temperature is $T_0$. The bubble rises to the surface where the absolute temperature is $2T_0$ and the pressure is half of the pressure at the bottom.
What is the final volume of the air bubble when it reaches the surface?

(A)

$V_0$

(B)

$2 V_0$

(C)

$4 V_0$

(D)

$8 V_0$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:Using ideal gas law $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$:
Given $P_1 = P_0, V_1 = V_0, T_1 = T_0$ and $P_2 = P_0/2, T_2 = 2T_0$.
$\frac{P_0 V_0}{T_0} = \frac{(P_0/2) V_2}{2 T_0} \implies 1 = \frac{V_2}{4 V_0} \implies V_2 = 4 V_0$.

Q14

Gas Type Degrees of Freedom ($f$) Ratio $\gamma = C_p/C_v$
Monoatomic 3 1.67
Diatomic (rigid) 5 1.40
Triatomic non-linear (rigid) 6 $\gamma_x$
What is the theoretical value of $\gamma_x$ for a non-linear rigid triatomic gas molecule?

Gas Type	Degrees of Freedom ($f$)	Ratio $\gamma = C_p/C_v$
Monoatomic	3	1.67
Diatomic (rigid)	5	1.40
Triatomic non-linear (rigid)	6	$\gamma_x$

(A)

1.67

(B)

1.40

(C)

1.33

(D)

1.25

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:For a rigid non-linear triatomic molecule, $f = 6$ (3 translational $+ 3$ rotational).
$C_v = \frac{f}{2}R = 3R$ and $C_p = C_v + R = 4R$.
Thus $\gamma = \frac{C_p}{C_v} = \frac{4R}{3R} = \frac{4}{3} \approx 1.33$.

Q15

At what temperature (in Kelvin) is the rms speed of Oxygen molecules ($O_2$, molar mass 32 g/mol) equal to the rms speed of Helium atoms (He, molar mass 4 g/mol) at 300 K?

(A)

600 K

(B)

1200 K

(C)

1800 K

(D)

2400 K

View Correct Answer & Detailed Solution

✓ Correct Option: (D)

Explanation:Equating rms speeds: $\sqrt{\frac{3RT_{O2}}{M_{O2}}} = \sqrt{\frac{3RT_{He}}{M_{He}}}$.
Squaring both sides: $\frac{T_{O2}}{32} = \frac{300}{4} \implies T_{O2} = 8 \times 300 = 2400\text{ K}$.

Q16

A rigid closed steel tank of constant volume contains an ideal gas at $27^\circ\text{C}$ (300 K) and initial pressure $P_0$. The tank is heated until its temperature reaches $327^\circ\text{C}$ (600 K).
What is the final pressure of the gas in terms of initial pressure $P_0$?

(A)

$1.5 P_0$

(B)

$2.0 P_0$

(C)

$3.0 P_0$

(D)

$4.0 P_0$

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation:At constant volume, Gay-Lussac's Law states $P \propto T$ (in Kelvin).
$T_1 = 27 + 273 = 300\text{ K}$ and $T_2 = 327 + 273 = 600\text{ K}$.
$\frac{P_2}{P_1} = \frac{600}{300} = 2 \implies P_2 = 2.0 P_0$.

Q17

What is the molar heat capacity at constant volume ($C_v$) for a rigid diatomic gas molecule with 5 degrees of freedom?

(A)

$\frac{3}{2} R$

(B)

$\frac{5}{2} R$

(C)

$\frac{7}{2} R$

(D)

$3 R$

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation:By the law of equipartition of energy, each degree of freedom contributes $\frac{1}{2} R$ per mole to internal energy.
For $f = 5$, internal energy per mole is $U = \frac{5}{2} R T$. Thus $C_v = \frac{dU}{dT} = \frac{5}{2} R$.

Q18

Gas molecules in a chamber have a mean free path $\lambda = 2.0 \times 10^{-7}m$ and an average speed $v_{avg} = 500m/s$.
What is the collision frequency ($f_{coll}$) of a molecule in this chamber?

(A)

$1.0 \times 10^8\text{ s}^{-1}$

(B)

$2.5 \times 10^8\text{ s}^{-1}$

(C)

$2.5 \times 10^9\text{ s}^{-1}$

(D)

$5.0 \times 10^9\text{ s}^{-1}$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:Collision frequency is $f_{coll} = \frac{v_{avg}}{\lambda}$.
$f_{coll} = \frac{500}{2.0 \times 10^{-7}} = 2.5 \times 10^9\text{ s}^{-1}$.

Q19

Statement I: According to the law of equipartition of energy, each quadratic degree of freedom contributes $\frac{1}{2} k_B T$ to the average energy of a particle.
Statement II: A solid containing $N_A$ atoms oscillating in 3 dimensions has 6 degrees of freedom per atom, leading to a molar specific heat capacity of $3R$.
Which of the following statements is correct?

(A)

Both Statement I and Statement II are true

(B)

Both Statement I and Statement II are false

(C)

Statement I is true, but Statement II is false

(D)

Statement I is false, but Statement II is true

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation:Statement I correctly defines equipartition of energy. Statement II correctly explains the Dulong-Petit Law ($C = 3R$) for solids, where each atom has 3 kinetic and 3 potential energy terms ($f = 6$). Both are true.

Q20

A container holds a mixture of 2 moles of Helium ($C_{v1} = 1.5 R$) and 3 moles of Nitrogen ($C_{v2} = 2.5 R$).
What is the effective molar heat capacity at constant volume ($C_{v,\text{mix}}$) of this mixture?

(A)

$1.8 R$

(B)

$2.0 R$

(C)

$2.1 R$

(D)

$2.3 R$

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation:$C_{v,\text{mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2}$.
Substituting values: $C_{v,\text{mix}} = \frac{2(1.5 R) + 3(2.5 R)}{2 + 3} = \frac{3 R + 7.5 R}{5} = \frac{10.5 R}{5} = 2.1 R$.

Grade 11 : Physics - Kinetic Theory (Set 4)

Grade 11 : Physics - Kinetic Theory (Set 4)— Questions & Detailed Solutions

Container geometry: A closed cubic vessel filled with an ideal gas of density $\rho = 1.2\text{ kg/m}^3$. The rms velocity of the gas molecules is measured as $v_{rms} = 500m/s$.
What is the pressure exerted by the gas molecules on the container walls?

A rigid cylinder contains 2 moles of Hydrogen gas ($H_2$) at temperature $T$. At this moderate temperature, vibrational modes are inactive, so each molecule possesses 5 active degrees of freedom.
What is the total internal energy ($U$) of the gas sample?

The mean free path of gas molecules in a chamber is $\lambda$. Due to chemical modification, the effective molecular diameter $d$ is doubled while keeping the number density $n$ of the molecules constant.
What is the new mean free path $\lambda'$ of the gas?

Gas Sample Gas Species Temperature ($K$) Molar Mass ($g/mol$)
Sample A Helium (He) 300 4
Sample B Neon (Ne) 300 20
Sample C Argon (Ar) 300 40
Based on the table above, which gas sample has the highest average translational kinetic energy per molecule?

Maxwell-Boltzmann velocity distribution curve: A plot showing three characteristic speeds on the horizontal axis ordered left-to-right as $v_1 < v_2 < v_3$.
Which set correctly identifies $v_1, v_2,$ and $v_3$?

What is the theoretical ratio of most probable speed ($v_{mp}$), average speed ($v_{avg}$), and root-mean-square speed ($v_{rms}$) for an ideal gas?

A thermal vessel contains a gas mixture of 1 mole of Helium (monoatomic, $f=3$) and 3 moles of Oxygen (diatomic rigid, $f=5$) in thermal equilibrium at temperature $T$.
What is the total internal energy of this gas mixture?

An ideal gas is sealed in a container of fixed volume $V$. The absolute temperature of the gas is increased from $T$ to $3T$.
By what factor does the rms speed of the gas molecules increase?

Statement I: The mean free path of gas molecules is inversely proportional to the mass density of the gas.
Statement II: At constant temperature, increasing the pressure of an ideal gas increases its mean free path.
Which of the following statements is correct?

What is the total translational kinetic energy of 1 mole of a monoatomic ideal gas at $T = 300\text{ K}$? (Take $R = 8.314\text{ J/(mol}\cdot\text{K)}$)

In the real gas van der Waals equation $\left(P + \frac{a}{V^2}\right)(V - b) = RT$, what physical correction does constant $a$ account for?

Gas Type Degrees of Freedom ($f$) Ratio $\gamma = C_p/C_v$
Monoatomic 3 1.67
Diatomic (rigid) 5 1.40
Triatomic non-linear (rigid) 6 $\gamma_x$
What is the theoretical value of $\gamma_x$ for a non-linear rigid triatomic gas molecule?

At what temperature (in Kelvin) is the rms speed of Oxygen molecules ($O_2$, molar mass 32 g/mol) equal to the rms speed of Helium atoms (He, molar mass 4 g/mol) at 300 K?

A rigid closed steel tank of constant volume contains an ideal gas at $27^\circ\text{C}$ (300 K) and initial pressure $P_0$. The tank is heated until its temperature reaches $327^\circ\text{C}$ (600 K).
What is the final pressure of the gas in terms of initial pressure $P_0$?

What is the molar heat capacity at constant volume ($C_v$) for a rigid diatomic gas molecule with 5 degrees of freedom?

Gas molecules in a chamber have a mean free path $\lambda = 2.0 \times 10^{-7}m$ and an average speed $v_{avg} = 500m/s$.
What is the collision frequency ($f_{coll}$) of a molecule in this chamber?

A container holds a mixture of 2 moles of Helium ($C_{v1} = 1.5 R$) and 3 moles of Nitrogen ($C_{v2} = 2.5 R$).
What is the effective molar heat capacity at constant volume ($C_{v,\text{mix}}$) of this mixture?

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Grade 11 : Physics - Kinetic Theory (Set 4)

Grade 11 : Physics - Kinetic Theory (Set 4)— Questions & Detailed Solutions

Container geometry: A closed cubic vessel filled with an ideal gas of density $\rho = 1.2\text{ kg/m}^3$. The rms velocity of the gas molecules is measured as $v_{rms} = 500m/s$.What is the pressure exerted by the gas molecules on the container walls?

A rigid cylinder contains 2 moles of Hydrogen gas ($H_2$) at temperature $T$. At this moderate temperature, vibrational modes are inactive, so each molecule possesses 5 active degrees of freedom.What is the total internal energy ($U$) of the gas sample?

The mean free path of gas molecules in a chamber is $\lambda$. Due to chemical modification, the effective molecular diameter $d$ is doubled while keeping the number density $n$ of the molecules constant.What is the new mean free path $\lambda'$ of the gas?

Gas SampleGas SpeciesTemperature ($K$)Molar Mass ($g/mol$)Sample AHelium (He)3004Sample BNeon (Ne)30020Sample CArgon (Ar)30040Based on the table above, which gas sample has the highest average translational kinetic energy per molecule?

Maxwell-Boltzmann velocity distribution curve: A plot showing three characteristic speeds on the horizontal axis ordered left-to-right as $v_1 < v_2 < v_3$.Which set correctly identifies $v_1, v_2,$ and $v_3$?

What is the theoretical ratio of most probable speed ($v_{mp}$), average speed ($v_{avg}$), and root-mean-square speed ($v_{rms}$) for an ideal gas?

A thermal vessel contains a gas mixture of 1 mole of Helium (monoatomic, $f=3$) and 3 moles of Oxygen (diatomic rigid, $f=5$) in thermal equilibrium at temperature $T$.What is the total internal energy of this gas mixture?

An ideal gas is sealed in a container of fixed volume $V$. The absolute temperature of the gas is increased from $T$ to $3T$.By what factor does the rms speed of the gas molecules increase?

Statement I: The mean free path of gas molecules is inversely proportional to the mass density of the gas.Statement II: At constant temperature, increasing the pressure of an ideal gas increases its mean free path.Which of the following statements is correct?

What is the total translational kinetic energy of 1 mole of a monoatomic ideal gas at $T = 300\text{ K}$? (Take $R = 8.314\text{ J/(mol}\cdot\text{K)}$)

In the real gas van der Waals equation $\left(P + \frac{a}{V^2}\right)(V - b) = RT$, what physical correction does constant $a$ account for?

Gas TypeDegrees of Freedom ($f$)Ratio $\gamma = C_p/C_v$Monoatomic31.67Diatomic (rigid)51.40Triatomic non-linear (rigid)6$\gamma_x$What is the theoretical value of $\gamma_x$ for a non-linear rigid triatomic gas molecule?

At what temperature (in Kelvin) is the rms speed of Oxygen molecules ($O_2$, molar mass 32 g/mol) equal to the rms speed of Helium atoms (He, molar mass 4 g/mol) at 300 K?

A rigid closed steel tank of constant volume contains an ideal gas at $27^\circ\text{C}$ (300 K) and initial pressure $P_0$. The tank is heated until its temperature reaches $327^\circ\text{C}$ (600 K).What is the final pressure of the gas in terms of initial pressure $P_0$?

What is the molar heat capacity at constant volume ($C_v$) for a rigid diatomic gas molecule with 5 degrees of freedom?

Gas molecules in a chamber have a mean free path $\lambda = 2.0 \times 10^{-7}m$ and an average speed $v_{avg} = 500m/s$.What is the collision frequency ($f_{coll}$) of a molecule in this chamber?

A container holds a mixture of 2 moles of Helium ($C_{v1} = 1.5 R$) and 3 moles of Nitrogen ($C_{v2} = 2.5 R$).What is the effective molar heat capacity at constant volume ($C_{v,\text{mix}}$) of this mixture?

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Container geometry: A closed cubic vessel filled with an ideal gas of density $\rho = 1.2\text{ kg/m}^3$. The rms velocity of the gas molecules is measured as $v_{rms} = 500m/s$.
What is the pressure exerted by the gas molecules on the container walls?

A rigid cylinder contains 2 moles of Hydrogen gas ($H_2$) at temperature $T$. At this moderate temperature, vibrational modes are inactive, so each molecule possesses 5 active degrees of freedom.
What is the total internal energy ($U$) of the gas sample?

The mean free path of gas molecules in a chamber is $\lambda$. Due to chemical modification, the effective molecular diameter $d$ is doubled while keeping the number density $n$ of the molecules constant.
What is the new mean free path $\lambda'$ of the gas?

Gas Sample Gas Species Temperature ($K$) Molar Mass ($g/mol$)
Sample A Helium (He) 300 4
Sample B Neon (Ne) 300 20
Sample C Argon (Ar) 300 40
Based on the table above, which gas sample has the highest average translational kinetic energy per molecule?

Maxwell-Boltzmann velocity distribution curve: A plot showing three characteristic speeds on the horizontal axis ordered left-to-right as $v_1 < v_2 < v_3$.
Which set correctly identifies $v_1, v_2,$ and $v_3$?

A thermal vessel contains a gas mixture of 1 mole of Helium (monoatomic, $f=3$) and 3 moles of Oxygen (diatomic rigid, $f=5$) in thermal equilibrium at temperature $T$.
What is the total internal energy of this gas mixture?

An ideal gas is sealed in a container of fixed volume $V$. The absolute temperature of the gas is increased from $T$ to $3T$.
By what factor does the rms speed of the gas molecules increase?

Statement I: The mean free path of gas molecules is inversely proportional to the mass density of the gas.
Statement II: At constant temperature, increasing the pressure of an ideal gas increases its mean free path.
Which of the following statements is correct?

Gas Type Degrees of Freedom ($f$) Ratio $\gamma = C_p/C_v$
Monoatomic 3 1.67
Diatomic (rigid) 5 1.40
Triatomic non-linear (rigid) 6 $\gamma_x$
What is the theoretical value of $\gamma_x$ for a non-linear rigid triatomic gas molecule?

A rigid closed steel tank of constant volume contains an ideal gas at $27^\circ\text{C}$ (300 K) and initial pressure $P_0$. The tank is heated until its temperature reaches $327^\circ\text{C}$ (600 K).
What is the final pressure of the gas in terms of initial pressure $P_0$?

Gas molecules in a chamber have a mean free path $\lambda = 2.0 \times 10^{-7}m$ and an average speed $v_{avg} = 500m/s$.
What is the collision frequency ($f_{coll}$) of a molecule in this chamber?

A container holds a mixture of 2 moles of Helium ($C_{v1} = 1.5 R$) and 3 moles of Nitrogen ($C_{v2} = 2.5 R$).
What is the effective molar heat capacity at constant volume ($C_{v,\text{mix}}$) of this mixture?