Ideal Gas Law and Kinetic Theory — Practice Quiz

A Physics cheat sheet for Ideal Gas Law and Kinetic Theory — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Ideal Gas Law: Relates pressure \(P\), volume \(V\), number of moles \(n\), and temperature \(T\).

$$PV = nRT$$

Where: \(R\) = universal gas constant \((8.31 J/(mol\cdot K))\)

Boyle's Law: For a fixed mass of ideal gas at constant temperature, pressure and volume are inversely proportional.

$$P_i V_i = P_f V_f$$

Charles' Law: For a fixed mass of ideal gas at constant pressure, volume is directly proportional to absolute temperature.

$$\frac{V_i}{T_i} = \frac{V_f}{T_f}$$

Average Kinetic Energy: Average translational kinetic energy of a gas molecule is proportional to absolute temperature.

$$\overline{KE} = \frac{1}{2} m v_{rms}^2 = \frac{3}{2} k T$$

Where: \(k\) = Boltzmann constant \((1.38 \times 10^{-23} J/K)\), \(v_{rms}\) = root-mean-square speed

Internal Energy (Monatomic Ideal Gas): Total internal energy \(U\) depends only on temperature.

$$U = \frac{3}{2} n R T$$

Fick's Law of Diffusion: Mass \(m\) diffusing in time \(t\) through area \(A\) and length \(L\) due to concentration difference \(\Delta C\).

$$m = \frac{(DA \Delta C) t}{L}$$

Where: \(D\) = diffusion constant

Practice quiz

  1. A sealed container holds an ideal gas at a certain pressure and temperature. If the volume of the container is halved while the temperature is kept constant, what happens to the pressure of the gas?

    • It doubles.
    • It halves.
    • It remains the same.
    • It quadruples.

    Answer: It doubles.

  2. A gas occupies $10 \text{ L}$ at a pressure of $2 \text{ atm}$. If the temperature remains constant, what volume will it occupy if the pressure is increased to $4 \text{ atm}$?

    • $2.5 \text{ L}$
    • $5 \text{ L}$
    • $10 \text{ L}$
    • $20 \text{ L}$

    Answer: $5 \text{ L}$

  3. A balloon contains $2.0 \text{ L}$ of air at $27 \text{ \textdegree C}$. If the pressure remains constant, what will be its volume if the temperature is increased to $127 \text{ \textdegree C}$?

    • $1.5 \text{ L}$
    • $2.0 \text{ L}$
    • $2.67 \text{ L}$
    • $3.0 \text{ L}$

    Answer: $2.67 \text{ L}$

  4. According to the kinetic theory of gases, if the absolute temperature of an ideal gas is doubled, what happens to the average translational kinetic energy of its molecules?

    • It remains the same.
    • It is halved.
    • It is doubled.
    • It is quadrupled.

    Answer: It is doubled.

  5. For a monatomic ideal gas, which of the following factors primarily determines its total internal energy $U$?

    • Pressure and volume.
    • Number of moles and temperature.
    • Volume and temperature.
    • Pressure and number of moles.

    Answer: Number of moles and temperature.

  6. Which of the following changes would increase the rate of diffusion of a substance according to Fick's Law?

    • Decreasing the diffusion constant $D$.
    • Increasing the length $L$ of the diffusion path.
    • Decreasing the concentration difference $\Delta C$.
    • Increasing the cross-sectional area $A$.

    Answer: Increasing the cross-sectional area $A$.

  7. An ideal gas is initially at pressure $P_1$, volume $V_1$, and temperature $T_1$. If its volume is compressed to $V_2 = \frac{1}{2} V_1$ and its temperature is increased to $T_2 = 2 T_1$, what is the new pressure $P_2$ in terms of $P_1$?

    • $P_2 = P_1$
    • $P_2 = 2 P_1$
    • $P_2 = 4 P_1$
    • $P_2 = \frac{1}{4} P_1$

    Answer: $P_2 = 4 P_1$

  8. What is the average translational kinetic energy of a gas molecule at $300 \text{ K}$? Use the Boltzmann constant $k = 1.38 \times 10^{-23} \text{ J/K}$.

    • $2.07 \times 10^{-21} \text{ J}$
    • $4.14 \times 10^{-21} \text{ J}$
    • $6.21 \times 10^{-21} \text{ J}$
    • $8.28 \times 10^{-21} \text{ J}$

    Answer: $6.21 \times 10^{-21} \text{ J}$

  9. Calculate the internal energy $U$ of $2.0$ moles of a monatomic ideal gas at a temperature of $400 \text{ K}$. Use the universal gas constant $R = 8.31 \text{ J/(mol} \cdot \text{K)}$.

    • $4986 \text{ J}$
    • $9972 \text{ J}$
    • $19944 \text{ J}$
    • $2493 \text{ J}$

    Answer: $9972 \text{ J}$

  10. A substance diffuses through a membrane with a diffusion constant $D = 2.0 \times 10^{-9} \text{ m}^2/\text{s}$. If the membrane has an area $A = 0.01 \text{ m}^2$ and length $L = 0.001 \text{ m}$, and the concentration difference is $\Delta C = 5.0 \text{ mol/m}^3$, what mass (in moles) diffuses in $100 \text{ s}$?

    • $1.0 \times 10^{-5} \text{ mol}$
    • $2.0 \times 10^{-5} \text{ mol}$
    • $5.0 \times 10^{-6} \text{ mol}$
    • $1.0 \times 10^{-6} \text{ mol}$

    Answer: $1.0 \times 10^{-5} \text{ mol}$

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