Ideal Gas Law and Kinetic Theory — Hard Practice Quiz

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

Practice quiz

1. A fixed amount of an ideal gas is contained in a rigid vessel. If the pressure of the gas is doubled, how does the root-mean-square speed of its molecules change?

   • It increases by a factor of $2$.
   • It increases by a factor of $\sqrt{2}$.
   • It remains unchanged.
   • It decreases by a factor of $\sqrt{2}$.

   Answer: It increases by a factor of $\sqrt{2}$.

2. A monatomic ideal gas undergoes an isothermal expansion. Which of the following statements is true regarding its internal energy and pressure?

   • Both internal energy and pressure remain constant.
   • Internal energy remains constant, and pressure decreases.
   • Internal energy increases, and pressure decreases.
   • Internal energy decreases, and pressure remains constant.

   Answer: Internal energy remains constant, and pressure decreases.

3. An ideal gas initially occupies a volume of $V_1$ at pressure $P_1$ and absolute temperature $T_1$. It is then compressed to a volume of $V_2 = V_1/3$ and simultaneously heated to an absolute temperature of $T_2 = 2T_1$. What is the final pressure $P_2$ in terms of $P_1$?

   • $P_2 = 6P_1$
   • $P_2 = \frac{2}{3} P_1$
   • $P_2 = \frac{3}{2} P_1$
   • $P_2 = 3P_1$

   Answer: $P_2 = 6P_1$

4. The diffusion constant $D$ for a gas is generally proportional to the average molecular speed, which in turn is proportional to the root-mean-square speed $v_{rms}$. If the absolute temperature of a gas is quadrupled, how would the diffusion constant $D$ for that gas change, assuming all other factors remain constant?

   • It increases by a factor of $2$.
   • It increases by a factor of $4$.
   • It increases by a factor of $16$.
   • It remains unchanged.

   Answer: It increases by a factor of $2$.

5. Two containers, A and B, hold monatomic ideal gases. Container A has $n$ moles at temperature $T$. Container B has $2n$ moles at temperature $T/2$. What is the ratio of the internal energy of gas A to gas B ($U_A/U_B$)?

   • $1:1$
   • $1:2$
   • $2:1$
   • $4:1$

   Answer: $1:1$

6. Derive an expression for the density $\rho$ of an ideal gas in terms of its pressure $P$, absolute temperature $T$, and molar mass $M$.

   • $\rho = \frac{PM}{RT}$
   • $\rho = \frac{RT}{PM}$
   • $\rho = \frac{P}{MRT}$
   • $\rho = \frac{M}{PRT}$

   Answer: $\rho = \frac{PM}{RT}$

7. A fixed amount of an ideal gas has its volume halved while its absolute temperature is doubled. How do the average kinetic energy of its molecules and the pressure of the gas change?

   • Average kinetic energy doubles, and pressure quadruples.
   • Average kinetic energy doubles, and pressure doubles.
   • Average kinetic energy remains constant, and pressure quadruples.
   • Average kinetic energy quadruples, and pressure doubles.

   Answer: Average kinetic energy doubles, and pressure quadruples.

8. According to Fick's Law, if the mass $m$ of a substance diffusing through a membrane is to be doubled in the same time $t$, by what factor must the concentration difference $\Delta C$ be changed, assuming the diffusion constant $D$, area $A$, and length $L$ remain constant?

   • It must be doubled.
   • It must be quadrupled.
   • It must be halved.
   • It must remain unchanged.

   Answer: It must be doubled.

9. For a fixed amount of a monatomic ideal gas, if the total internal energy $U$ is doubled, how does the root-mean-square speed $v_{rms}$ of its molecules change?

   • It increases by a factor of $2$.
   • It increases by a factor of $\sqrt{2}$.
   • It remains unchanged.
   • It decreases by a factor of $\sqrt{2}$.

   Answer: It increases by a factor of $\sqrt{2}$.

10. A fixed amount of ideal gas initially at pressure $P_1$, volume $V_1$, and absolute temperature $T_1$ undergoes a two-step process. First, it is compressed isothermally to a volume $V_2 = V_1/2$. Second, its temperature is then raised to $T_2 = 2T_1$ at constant volume $V_2$. What is the final pressure $P_f$ of the gas?

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

    Answer: $P_f = 4P_1$