Practice Questions

Recall the expression for the pressure exerted by an ideal gas, $P = \frac{1}{3} n m \overline{v^2}$. Identify what each symbol ($n, m, \overline{v^2}$) represents.

Justify the statement: 'At the same temperature, a molecule of hydrogen gas has a higher root mean square speed than a molecule of oxygen gas.' Use the kinetic interpretation of temperature to support your argument.

Analyze what happens to the average translational kinetic energy of molecules in an ideal gas if its absolute temperature is quadrupled.

State the relationship between the universal gas constant ($R$), Avogadro's number ($N_A$), and Boltzmann's constant ($k_B$).

Recall the ideal gas equation in terms of the number of moles ($\mu$) and the universal gas constant ($R$).

Calculate the pressure exerted by $2$ moles of an ideal gas kept in a container of volume $0.05 \text{ m}^3$ at a temperature of $300 \text{ K}$. Use the ideal gas equation and assume $R = 8.314 \text{ J mol}^{-1} \text{K}^{-1}$.

State Avogadro's law regarding the volume and number of molecules of gases.

Justify why the average kinetic energy of gas molecules is considered a measure of temperature, but the average momentum is not.

Define the term 'mean free path' as used in the kinetic theory of gases.

Name the physical quantity that is a measure of the average translational kinetic energy of the molecules of a gas.

Apply the law of equipartition of energy to determine the average energy associated with a single translational degree of freedom of a gas molecule at temperature T.

Identify the factors on which the mean free path of a gas molecule depends and describe the relationship.

List the four main assumptions of the kinetic theory of an ideal gas.

Describe how the pressure of a gas is explained by the kinetic theory.

An ideal gas has its temperature changed from $27^{\circ}\text{C}$ to $327^{\circ}\text{C}$. Recall the formula for the root mean square speed and find the ratio of its initial rms speed to its final rms speed.

A container holds a mixture of helium (He) and nitrogen ($N_2$) gas at the same temperature. Compare the root mean square ($v_{rms}$) speeds of the molecules of the two gases. The atomic mass of He is $4.0 \text{ u}$ and the molecular mass of $N_2$ is $28.0 \text{ u}$.

A cube of side $10 \text{ cm}$ contains helium gas. The number of molecules per unit volume is $2 \times 10^{25} \text{ m}^{-3}$ and the mass of a helium atom is $6.64 \times 10^{-27} \text{ kg}$. If the root mean square speed of the atoms is $1360 \text{ m/s}$, solve for the pressure inside the cube.

Examine the relationship $PV = \frac{2}{3} E$, where $E$ is the total translational kinetic energy of an ideal gas. How does this equation connect the macroscopic properties ($P, V$) of a gas to its microscopic property ($E$)?

Contrast the behavior of a real gas with that of an ideal gas, particularly under conditions of high pressure and low temperature. Explain the reasons for the deviations based on the assumptions of the kinetic theory.

Evaluate the assumption in the kinetic theory that molecular collisions are perfectly elastic. Why is this assumption crucial for the derivation of the ideal gas pressure equation, $P = \frac{1}{3} n m \overline{v^2}$?

What is the value of the molar specific heat at constant volume ($C_V$) for a monatomic ideal gas in terms of the universal gas constant $R$?

Explain the kinetic interpretation of temperature based on the kinetic theory of gases.

Explain Dalton's law of partial pressures using the kinetic theory of gases.

A cylinder with a fixed volume contains $3$ moles of a monatomic ideal gas at an initial temperature of $27^{\circ}\text{C}$. Calculate the amount of heat required to raise the temperature of the gas to $127^{\circ}\text{C}$. Also, calculate the change in its internal energy. (Given $R = 8.31 \text{ J mol}^{-1} \text{K}^{-1}$)

Analyze how the mean free path of gas molecules in a container changes if the gas is compressed to half its original volume while keeping the temperature constant.

Compare the value of the ratio of specific heats ($\gamma = C_p/C_v$) for a monatomic gas and a rigid diatomic gas. Analyze the reason for the difference based on their degrees of freedom.

Calculate the root mean square speed of an oxygen molecule ($O_2$) at a temperature of $27^{\circ}\text{C}$. The molecular mass of oxygen is $32.0 \text{ g/mol}$, and $R = 8.314 \text{ J mol}^{-1} \text{K}^{-1}$.

Formulate a model to explain why the specific heat capacity of a diatomic gas increases at very high temperatures. Your model should critique the 'rigid rotator' assumption and incorporate vibrational degrees of freedom.

Critique the ideal gas model. Under what conditions does it fail to accurately predict the behavior of a real gas? Propose two main reasons for its failure, relating them to the postulates of the kinetic theory.

Propose a method to increase the mean free path of gas molecules within a sealed container without changing the type of gas or the container's volume. Justify your proposal using the formula for mean free path, $l = \frac{1}{\sqrt{2} n \pi d^2}$.

Propose a reason why the law of equipartition of energy fails to explain the specific heats of gases at very low temperatures.

Create a scenario where two different ideal gases, Helium (He) and Argon (Ar), are mixed in a container. If the ratio of their masses is $1:10$ and the temperature is constant at $300 \text{ K}$, formulate an expression for the ratio of their root mean square speeds, $(v_{rms})_{He} / (v_{rms})_{Ar}$.

Justify why, for a polyatomic molecule with $f$ vibrational modes, the molar specific heat at constant volume is given by $C_V = (3+f)R$, assuming 3 translational and 3 rotational degrees of freedom.

Calculate the temperature at which the root mean square speed of an argon atom (Ar) is equal to the rms speed of a neon atom (Ne) at $-73^{\circ}\text{C}$. The atomic mass of Ar is $39.9 \text{ u}$ and that of Ne is $20.2 \text{ u}$.

Explain the concept of degrees of freedom and list the number of translational and rotational degrees of freedom for monatomic, diatomic (rigid), and non-linear polyatomic gases.

Summarize the law of equipartition of energy and explain how it applies to translational, rotational, and vibrational degrees of freedom for a molecule.

Design a hypothetical experiment to verify the law of equipartition of energy for a diatomic gas like Nitrogen ($N_2$) at moderate temperatures. Your design should specify the measurements needed and how you would calculate the molar specific heats ($C_V$ and $C_p$) to justify that it has 5 degrees of freedom.

Evaluate the claim that the total pressure of a mixture of non-reacting gases is the sum of their partial pressures (Dalton's Law). Justify this law starting from the kinetic theory pressure equation $P = \frac{1}{3} n m \overline{v^2}$ and the kinetic interpretation of temperature.

Design an experiment to estimate the diameter of an argon atom using the concept of mean free path. You are given a container of argon gas at a known pressure $P$ and temperature $T$, and you can measure the gas's viscosity, which is related to the mean free path. Outline the steps and the formulas you would use to arrive at the diameter $d$.

Critique the assumption that molecules in an ideal gas only collide with the walls and not with each other. Explain why, despite this simplification, the final expression $P=(1/3)nm\overline{v^2}$ remains valid for a gas in a steady state.

A scientist claims to have created a gas where all molecules move at the exact same speed. Evaluate this claim in the context of the kinetic theory of gases. Would the concepts of 'temperature' and 'pressure' as derived from kinetic theory still be valid for this hypothetical gas? Justify your reasoning.

A vessel is divided into two equal compartments by a partition. One compartment contains Helium (monatomic) and the other contains Oxygen (diatomic), both at the same initial temperature $T$ and pressure $P$. Formulate a prediction for what happens to the total pressure and the temperature of the mixture if the partition is removed. Justify your prediction.

Demonstrate that for a mixture of non-reacting ideal gases, the total pressure is the sum of the partial pressures exerted by each gas (Dalton's Law), using the kinetic theory of gases.

A vessel of volume $V$ contains a mixture of two non-reactive gases, Hydrogen ($H_2$) and Oxygen ($O_2$), at temperature $T$. The partial pressure of Hydrogen is twice that of Oxygen. The molecular mass of $H_2$ is $2.0 \text{ u}$ and that of $O_2$ is $32.0 \text{ u}$. Calculate the ratio of (i) the number of molecules of $H_2$ to $O_2$ and (ii) the mass density of $H_2$ to $O_2$ in the mixture.

Calculate the mean free path for a nitrogen molecule ($N_2$) in a sample of gas at standard temperature and pressure (STP). At STP, the number density of molecules is approximately $2.7 \times 10^{25} \text{ m}^{-3}$. Take the diameter of a nitrogen molecule to be $3.15 \times 10^{-10} \text{ m}$.