The Ideal Gas Law - AP Physics 2

1. A sample of ideal gas is trapped in a rigid, sealed container (constant volume and constant amount of gas). The initial state is $P_1=150\ \text{kPa}$ at $T_1=300\ \text{K}$. The container is heated to $T_2=600\ \text{K}$ with no leaks. Which statement correctly describes the final pressure $P_2$?
   Answer: $P_2=300\ \text{kPa}$ because pressure is proportional to temperature in kelvins This question tests the ideal gas law. For an ideal gas, PV = nRT relates pressure, volume, temperature, and amount of gas. In a rigid container with constant volume and constant amount of gas, we can use Gay-Lussac's Law: P₁/T₁ = P₂/T₂. Substituting the values: 150 kPa / 300 K = P₂ / 600 K, which gives P₂ = 150 kPa × (600 K / 300 K) = 300 kPa. Choice C incorrectly assumes pressure is inversely proportional to temperature, showing confusion about which variables are held constant. When volume and amount are constant, use P₁/T₁ = P₂/T₂ with absolute temperatures.
2. An ideal gas is in a piston-cylinder at $P_1=200\ \text{kPa}$, $V_1=1.5\ \text{L}$, and $T=300\ \text{K}$. The amount of gas and temperature are held constant as the piston moves to $V_2=3.0\ \text{L}$. Which statement correctly describes the final pressure?
   Answer: $P_2=100\ \text{kPa}$, because doubling volume halves pressure at constant $T$ This problem tests understanding of the ideal gas law. The ideal gas law relates pressure, volume, temperature, and amount of gas through PV = nRT. When temperature and amount of gas are constant, Boyle's Law applies: P₁V₁ = P₂V₂. Substituting the given values: (200 kPa)(1.5 L) = P₂(3.0 L), which gives P₂ = 100 kPa. Choice B incorrectly assumes pressure increases with volume, when actually they are inversely proportional at constant temperature. To solve ideal gas problems, first identify which variables are constant, then apply the appropriate simplified form of the ideal gas law.
3. A sealed syringe contains ideal gas at temperature $T$ and volume $V$. The plunger is pushed in so the volume becomes $0.80V$ while temperature and amount of gas remain constant. Which statement correctly describes the final pressure $P_2$ compared to $P_1$?
   Answer: $P_2=1.25P_1$ This problem tests understanding of the ideal gas law. The ideal gas law PV = nRT connects pressure, volume, temperature, and the amount of gas present. When temperature and amount of gas are constant, pressure and volume are inversely proportional: P₁V₁ = P₂V₂. If volume decreases to 0.80V, pressure increases by the reciprocal factor: P₂ = P₁ × (V₁/0.80V₁) = P₁/0.80 = 1.25P₁. Choice A incorrectly calculates P₂ = 0.80P₁, confusing the direct relationship with volume change for an inverse relationship. To avoid this error, remember that in isothermal processes, pressure and volume change in opposite directions.
4. An ideal gas in a cylinder with a movable piston is compressed slowly from $4.0\,\text{L}$ to $2.0\,\text{L}$ while the temperature and amount of gas are held constant. Initially the pressure is $100\,\text{kPa}$. Compared to the initial pressure, the final pressure is
   Answer: $200\,\text{kPa}$ because pressure is inversely proportional to volume. This question tests the ideal gas law. The ideal gas law (PV = nRT) shows that when temperature and amount of gas are constant, pressure and volume are inversely proportional (P₁V₁ = P₂V₂). As volume decreases from 4.0 L to 2.0 L (halved), pressure must double to maintain the same product PV. Therefore, pressure increases from 100 kPa to 200 kPa. Choice A incorrectly states that pressure decreases when volume decreases, revealing a fundamental misconception about the inverse relationship between P and V. To solve ideal gas problems correctly, identify which variables are constant and apply the appropriate proportionality relationship.
5. An ideal gas occupies volume $V$ at pressure $P$ and temperature $T$. The temperature is doubled to $2T$ while pressure is held constant and no gas is added or removed. Which statement correctly describes the new volume?
   Answer: $V_2=2V_1$ This problem tests understanding of the ideal gas law. The ideal gas law PV = nRT shows that pressure, volume, temperature, and amount of gas are all related. When pressure and amount of gas remain constant, volume is directly proportional to absolute temperature: V₁/T₁ = V₂/T₂. If temperature doubles from T to 2T, volume also doubles: V₂ = V₁ × (2T/T) = 2V₁. Choice D incorrectly attempts to subtract 273 from the temperature ratio, showing confusion between Kelvin conversion and temperature ratios. When using the ideal gas law, work with temperature ratios directly if temperatures are already in Kelvin.
6. A sealed syringe contains an ideal gas at $P_1=80\ \text{kPa}$, $V_1=50\ \text{mL}$, and $T_1=300\ \text{K}$. The amount of gas stays constant. The syringe is pushed to $V_2=25\ \text{mL}$ while kept at $T_2=300\ \text{K}$. Which statement correctly describes $P_2$?
   Answer: $P_2=160\ \text{kPa}$, because halving volume doubles pressure at constant $T$ This problem tests understanding of the ideal gas law. At constant temperature with a fixed amount of gas, Boyle's Law applies: P₁V₁ = P₂V₂. The volume is halved from 50 mL to 25 mL, so the pressure must double to maintain the constant product. Therefore, P₂ = P₁ × (V₁/V₂) = 80 kPa × (50 mL/25 mL) = 160 kPa. Choice A incorrectly suggests pressure decreases when volume decreases, which contradicts the inverse relationship. For isothermal processes with ideal gases, remember that pressure and volume are inversely proportional—halving the volume doubles the pressure.
7. A container of ideal gas is kept at constant temperature $T$. The pressure is increased from $P_1$ to $3P_1$ while temperature and amount of gas remain constant. Which statement correctly describes the final volume $V_2$ compared to $V_1$?
   Answer: $V_2=\tfrac{1}{3}V_1$ This problem tests understanding of the ideal gas law. The ideal gas law PV = nRT describes how pressure, volume, temperature, and amount of gas are interdependent. At constant temperature and amount of gas, pressure and volume are inversely proportional: P₁V₁ = P₂V₂. When pressure triples from P₁ to 3P₁, volume must decrease to one-third: V₂ = V₁ × (P₁/3P₁) = ⅓V₁. Choice A incorrectly suggests volume triples when pressure triples, missing the inverse relationship in Boyle's Law. Always identify which variables are held constant and apply the correct proportional relationship between the changing variables.
8. An ideal gas in a cylinder is kept at constant pressure by a movable piston. Initially $V_1=4.0\,\text{L}$ at $T_1=250\,\text{K}$. It is warmed to $T_2=300\,\text{K}$ while pressure and amount of gas remain constant. Which statement correctly describes $V_2$?
   Answer: It is $4.8\,\text{L}$. This question tests the ideal gas law. The ideal gas law PV = nRT shows how pressure, volume, temperature, and moles are interconnected. When pressure and moles are constant, volume is directly proportional to temperature: V₁/T₁ = V₂/T₂. With V₁ = 4.0 L, T₁ = 250 K, and T₂ = 300 K, we calculate V₂ = V₁T₂/T₁ = (4.0 L)(300 K)/(250 K) = 4.8 L. Choice D incorrectly uses the temperature difference in Celsius (50°C) rather than the ratio of absolute temperatures. Always convert to Kelvin and use ratios when applying Charles's Law: V₁/T₁ = V₂/T₂.
9. A fixed amount of ideal gas undergoes a change where the temperature is held constant and the pressure triples. No gas enters or leaves the container. Which statement correctly describes the final volume compared to the initial volume?
   Answer: The final volume is one-third the initial volume This question tests the ideal gas law. For a fixed amount of gas at constant temperature, the ideal gas law PV = nRT reduces to Boyle's Law: P₁V₁ = P₂V₂, showing that pressure and volume are inversely proportional. If pressure triples (P₂ = 3P₁), then volume must be reduced to one-third: V₂ = V₁/3. Choice C incorrectly claims volume is unchanged because temperature is constant, missing that P and V can still vary inversely at constant T. When temperature and amount are constant, remember that PV remains constant—if one variable increases, the other must decrease proportionally.
10. A fixed-volume container holds an ideal gas at $27^\circ\text{C}$ and pressure $P_1$. It is heated to $127^\circ\text{C}$ while volume and amount of gas remain constant. Which statement correctly describes the new pressure $P_2$?
    Answer: $P_2=\tfrac{400}{300}P_1$ This problem tests understanding of the ideal gas law. The ideal gas law PV = nRT shows how pressure, volume, temperature, and amount of gas are related. For constant volume and amount, pressure is directly proportional to absolute temperature: P₁/T₁ = P₂/T₂. Converting temperatures to Kelvin: T₁ = 27°C + 273 = 300 K and T₂ = 127°C + 273 = 400 K, so P₂ = P₁ × (400/300) = (4/3)P₁. Choice A incorrectly uses Celsius temperatures (127/27), demonstrating the critical misconception of not converting to absolute temperature. Always convert Celsius to Kelvin before applying gas law relationships.
11. A balloon contains an ideal gas at $P=1.0\ \text{atm}$ and $T=300\ \text{K}$. The external pressure stays at $1.0\ \text{atm}$, so the gas pressure is constant. If the number of moles inside doubles while temperature remains constant, which statement correctly describes the balloon's volume?
    Answer: The volume doubles, because $V$ is proportional to $n$ at constant $P$ and $T$ This problem tests understanding of the ideal gas law. At constant pressure and temperature, the ideal gas law shows that volume is directly proportional to the number of moles: V ∝ n. Since the balloon maintains equilibrium with external pressure (1.0 atm), its internal pressure stays constant. When the number of moles doubles while P and T remain constant, the volume must also double to satisfy PV = nRT. Choice C incorrectly assumes adding gas shrinks the balloon, misunderstanding that constant pressure allows expansion. For problems involving flexible containers like balloons, recognize that pressure equilibrates with surroundings, making volume proportional to the amount of gas.
12. An ideal gas is sealed in a syringe so the amount of gas stays constant. The gas is compressed slowly while kept in thermal contact with a large water bath at $T=300\ \text{K}$ (temperature constant). The volume changes from $V_1=40\ \text{mL}$ to $V_2=10\ \text{mL}$. Which statement correctly describes the pressure change?
    Answer: The final pressure is four times the initial pressure This question tests the ideal gas law. The ideal gas law PV = nRT shows how pressure, volume, temperature, and amount of gas are related. For constant temperature and constant amount of gas, we use Boyle's Law: P₁V₁ = P₂V₂. Since volume decreases from 40 mL to 10 mL (a factor of 4), pressure must increase by the same factor to maintain the product PV constant. Choice D incorrectly focuses on the 30 mL difference rather than the ratio, demonstrating a misconception about how to apply proportional relationships. When temperature and amount are constant, use P₁V₁ = P₂V₂ and work with ratios.
13. A fixed amount of ideal gas is sealed in a cylinder with a movable piston. Initially $P_1=200\ \text{kPa}$, $V_1=3.0\ \text{L}$, and $T_1=300\ \text{K}$. The gas is slowly heated while the piston moves so the pressure is held constant at $200\ \text{kPa}$ and no gas enters or leaves. When the temperature reaches $T_2=450\ \text{K}$, which statement correctly describes the final volume $V_2$?
    Answer: $V_2=4.5\ \text{L}$ because volume is proportional to temperature in kelvins This question tests the ideal gas law. The ideal gas law states that PV = nRT, where pressure (P), volume (V), and temperature (T) are related for a fixed amount of gas (n moles). Since the pressure is held constant and the amount of gas is fixed, we can use Charles's Law: V₁/T₁ = V₂/T₂. Substituting the given values: 3.0 L / 300 K = V₂ / 450 K, which gives V₂ = 3.0 L × (450 K / 300 K) = 4.5 L. Choice C incorrectly uses Celsius temperatures without converting to Kelvin, which would give 6.75 L—this is a common misconception that temperature ratios work with any scale. When pressure and amount are constant, always use absolute temperature (Kelvin) and apply V₁/T₁ = V₂/T₂.
14. A rigid, sealed steel tank contains $1.0\ \text{mol}$ of an ideal gas at $300\ \text{K}$ and pressure $P_1$. The tank is heated to $450\ \text{K}$ while volume and amount of gas remain constant. Which statement correctly describes the new pressure $P_2$ compared to $P_1$?
    Answer: $P_2=1.5P_1$ This problem tests understanding of the ideal gas law. The ideal gas law states that PV = nRT, where pressure (P), volume (V), and temperature (T) are related through the number of moles (n) and gas constant (R). When volume and amount of gas are constant, pressure is directly proportional to absolute temperature: P₁/T₁ = P₂/T₂. Since temperature increases from 300 K to 450 K, we get P₂ = P₁ × (450/300) = 1.5P₁. Choice D incorrectly uses Celsius temperature (450/(300+273)), showing the common misconception of not recognizing that temperatures are already in Kelvin. Always verify that temperatures are in absolute units (Kelvin) and identify which variables remain constant to determine the appropriate gas law relationship.
15. A container of fixed volume holds an ideal gas at $P_1=100\,\text{kPa}$ and $T_1=400\,\text{K}$. Some gas escapes through a valve until the amount of gas is halved, while the temperature is maintained at $400\,\text{K}$. Which statement correctly describes the final pressure $P_2$?
    Answer: $P_2$ is $50\,\text{kPa}$ because pressure is proportional to the amount of gas at fixed $V$ and $T$. This question tests understanding of the ideal gas law. The ideal gas law PV = nRT shows that pressure is directly proportional to the amount of gas (n) when volume and temperature are constant. If half the gas escapes, n₂ = n₁/2, so the pressure becomes half: P₂ = P₁ × (n₂/n₁) = 100 kPa × (1/2) = 50 kPa. Choice A incorrectly claims fewer moles means higher pressure, revealing a fundamental misconception about the direct relationship between pressure and amount of gas. To solve gas law problems, identify which variables change and which remain constant, then use the appropriate proportionality relationships.
16. A sealed syringe contains ideal gas at $P_1=1.0\,\text{atm}$, $V_1=20\,\text{mL}$, and $T_1=300\,\text{K}$. The gas is cooled to $T_2=150\,\text{K}$ while the plunger is held fixed so the volume remains $20\,\text{mL}$. Which statement correctly describes the final pressure $P_2$?
    Answer: $P_2$ is $0.50\,\text{atm}$ because pressure is proportional to absolute temperature at constant volume. This question tests understanding of the ideal gas law. When volume and amount of gas are constant (sealed syringe with fixed plunger), pressure is directly proportional to absolute temperature: P₁/T₁ = P₂/T₂. Since T₂ = 150 K is half of T₁ = 300 K, the pressure also becomes half: P₂ = P₁ × (T₂/T₁) = 1.0 atm × (150 K / 300 K) = 0.50 atm. Choice A incorrectly claims lower temperature increases pressure, showing a fundamental misconception about the direct relationship between P and T. When volume is constant, use Gay-Lussac's Law (P/T = constant) and remember that pressure decreases with temperature.
17. An ideal gas in a sealed syringe is warmed while the plunger is held fixed, so volume and amount of gas are constant. The pressure rises from $80\,\text{kPa}$ to $120\,\text{kPa}$. Which statement correctly describes the temperature change?
    Answer: It increases by a factor of $1.5$ because temperature is proportional to pressure (in kelvins). This question tests the ideal gas law. When volume and amount of gas are constant, the ideal gas law (PV = nRT) shows that pressure is directly proportional to absolute temperature (P₁/T₁ = P₂/T₂). Since pressure increases from 80 kPa to 120 kPa (a factor of 1.5), temperature must also increase by a factor of 1.5. Choice C incorrectly calculates the temperature change as 120 - 80 = 40 K, confusing the pressure difference with temperature change and failing to recognize the proportional relationship. To find temperature changes in ideal gas problems, use the ratio of pressures to find the ratio of absolute temperatures when V and n are constant.
18. A sealed $1.0\,\text{L}$ flask contains ideal gas at $1.2\,\text{atm}$ and $300\,\text{K}$. The flask is heated to $360\,\text{K}$ while the volume and amount of gas are held constant. The final pressure is closest to
    Answer: $1.44\,\text{atm}$ because $P\propto T$ in kelvins at constant $V$ and $n$. This question tests the ideal gas law. In a sealed flask with constant volume and amount of gas, pressure is directly proportional to absolute temperature (P₁/T₁ = P₂/T₂). The temperature increases from 300 K to 360 K, a factor of 360/300 = 1.2. Therefore, pressure increases by the same factor: 1.2 atm × 1.2 = 1.44 atm. Choice D incorrectly adds the temperature difference (60 K) scaled by some factor to the initial pressure, showing confusion between proportional relationships and arithmetic operations. When applying the ideal gas law, use multiplicative factors based on temperature ratios, not additive differences.
19. A fixed amount of ideal gas is taken from state 1 to state 2 in a cylinder. The temperature is held constant at $300\,\text{K}$ while the volume increases from $1.5\,\text{L}$ to $3.0\,\text{L}$. If $P_1=300\,\text{kPa}$, then $P_2$ is
    Answer: $150\,\text{kPa}$ because pressure is inversely proportional to volume. This question tests the ideal gas law. At constant temperature and amount of gas, the ideal gas law (PV = nRT) reduces to Boyle's Law: P₁V₁ = P₂V₂, showing that pressure and volume are inversely proportional. When volume doubles from 1.5 L to 3.0 L, pressure must be halved to maintain the constant product PV. Therefore, pressure decreases from 300 kPa to 150 kPa. Choice A incorrectly states that pressure increases when volume increases, revealing a fundamental misconception about the inverse relationship between P and V at constant temperature. When temperature is constant, remember that pressure and volume change in opposite directions by inverse factors.
20. A flexible balloon contains ideal gas at constant temperature and external pressure. The balloon initially has $n_1=0.20\,\text{mol}$ and volume $V_1=1.0\,\text{L}$. Gas is added to reach $n_2=0.30\,\text{mol}$ while temperature and pressure remain constant. Compared to $V_1$, what is $V_2$?
    Answer: It is $1.5\,\text{L}$. This question tests the ideal gas law. According to PV = nRT, pressure, volume, temperature, and moles of gas are related in specific ways. When pressure and temperature are constant, volume is directly proportional to the number of moles: V₁/n₁ = V₂/n₂. With V₁ = 1.0 L, n₁ = 0.20 mol, and n₂ = 0.30 mol, we find V₂ = V₁n₂/n₁ = (1.0 L)(0.30 mol)/(0.20 mol) = 1.5 L. Choice D incorrectly suggests using Celsius temperature, which is irrelevant when temperature is constant. When P and T are constant, use Avogadro's Law: V₁/n₁ = V₂/n₂.

Start this quiz