Gas Laws and Kinetic Molecular Theory (4B)
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MCAT Chemical and Physical Foundations of Biological Systems › Gas Laws and Kinetic Molecular Theory (4B)
In a respiratory physiology experiment, an alveolar gas sample is approximated as an ideal gas at constant temperature. The partial pressure of $\mathrm{O_2}$ is measured as $P_{\mathrm{O_2}}=100\ \mathrm{mmHg}$ in a mixture with total pressure $P_{\text{total}}=760\ \mathrm{mmHg}$. Assume the mixture behaves ideally.
Based on Dalton’s law of partial pressures, which value is most consistent with the mole fraction of $\mathrm{O_2}$ in the sample?
$x_{\mathrm{O_2}}\approx 0.76$
$x_{\mathrm{O_2}}\approx 7.6$
$x_{\mathrm{O_2}}\approx 0.13$
$x_{\mathrm{O_2}}\approx 0.013$
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of Dalton's law and mole fractions. Dalton's law states that in a gas mixture, each component's partial pressure equals its mole fraction times total pressure: Pi = xiPtotal, which rearranges to xi = Pi/Ptotal. In this scenario, oxygen has partial pressure 100 mmHg in a mixture with total pressure 760 mmHg. The correct choice, A, follows because xO₂ = PO₂/Ptotal = 100/760 ≈ 0.13, representing oxygen as about 13% of the gas mixture by moles. Choice C is incorrect because it appears to calculate 760-100 = 660 then divide by total pressure, which has no physical meaning for mole fraction. In similar questions, remember that mole fractions must sum to 1.0 for all components and individual mole fractions must be between 0 and 1.
A sample of an ideal gas is confined in a frictionless piston-cylinder assembly at constant temperature (isothermal). The gas is compressed quasi-statically from $2.0\ \mathrm{L}$ to $1.0\ \mathrm{L}$ with no change in moles of gas.
Based on Boyle’s law, what effect would this compression most likely have on the gas pressure?
The pressure would decrease because compression reduces the frequency of wall collisions.
The pressure would approximately halve because $P\propto V$ at constant $T$ and $n$.
The pressure would remain unchanged because temperature is constant and pressure depends only on $T$.
The pressure would approximately double because $P\propto 1/V$ at constant $T$ and $n$.
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of isothermal compression using Boyle's law. Boyle's law states that for a fixed amount of gas at constant temperature, pressure is inversely proportional to volume (P ∝ 1/V or PV = constant). In this scenario, the isothermal compression from 2.0 L to 1.0 L represents a halving of volume while maintaining constant T and n. The correct choice, A, follows because when volume is halved at constant T and n, pressure must double to maintain the constant PV product required by Boyle's law. Choice B is incorrect because it suggests direct proportionality between P and V, which contradicts the inverse relationship. In similar questions, remember that isothermal processes maintain constant temperature, and for ideal gases, this means PV remains constant throughout the process.
A gas mixture is held at constant external pressure in a calibrated cylinder with a freely moving piston. The sample is heated from $300\ \mathrm{K}$ to $450\ \mathrm{K}$ while maintaining the same number of moles.
Based on Charles’s law, which prediction is most consistent with the gas volume change?
The volume should increase by a factor of $1.5$ because $V\propto T$ at constant $P$ and $n$.
The volume should remain constant because the piston prevents changes in volume at constant pressure.
The volume should decrease by a factor of $1.5$ because increasing $T$ increases gas density at constant $P$.
The volume should increase by a factor of $2.0$ because $V\propto T^2$ for an ideal gas at constant $P$.
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of isobaric heating using Charles's law. Charles's law states that for a fixed amount of gas at constant pressure, volume is directly proportional to absolute temperature (V ∝ T or V/T = constant). In this scenario, heating from 300 K to 450 K represents a temperature increase by a factor of 450/300 = 1.5 while maintaining constant pressure and moles. The correct choice, B, follows because at constant P and n, volume must increase by the same factor as temperature (1.5) to maintain the constant V/T ratio required by Charles's law. Choice D is incorrect because it suggests V ∝ T², which has no basis in ideal gas behavior. In similar questions, calculate the ratio of final to initial temperatures to determine the volume change factor for isobaric processes.
A physical chemistry lab compares predicted vs. observed behavior for $\mathrm{CO_2}$ in a rigid container at $T=300\ \mathrm{K}$ and high pressure. The ideal-gas model predicts $PV=nRT$ with $R=0.082\ \mathrm{L,atm,mol^{-1},K^{-1}}$. Under the same $n$, $V$, and $T$, the measured pressure is higher than the ideal-gas prediction.
Which interpretation is most consistent with real-gas effects at high pressure?
At high pressure, attractive intermolecular forces dominate, always lowering measured pressure relative to ideal behavior.
At high pressure, finite molecular volume becomes significant, effectively increasing measured pressure relative to ideal behavior.
At high pressure, the ideal gas law overestimates pressure because gases become perfectly incompressible.
At high pressure, the gas must obey Boyle’s law exactly, so any deviation indicates a temperature change.
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of real gas deviations at high pressure. Real gases deviate from ideal behavior due to two main factors: finite molecular volume and intermolecular forces. In this scenario, CO₂ at high pressure shows measured pressure higher than ideal predictions, indicating the dominant effect of finite molecular volume. The correct choice, A, follows because at high pressure, molecules are compressed closer together, and their finite volume becomes significant, effectively reducing the available free volume and causing measured pressure to exceed ideal predictions. Choice B is incorrect because while attractive forces do exist, they would lower (not raise) the measured pressure relative to ideal behavior. In similar questions, remember that at high pressure, volume effects typically dominate, while at moderate pressure and low temperature, attractive forces become more significant.
A sealed vessel contains $\mathrm{He}$ at $T=350\ \mathrm{K}$. The kinetic molecular theory model is used to compare the average molecular speed before and after changing temperature. Assume ideal-gas behavior and that only temperature changes.
Which prediction aligns with kinetic molecular theory regarding the average molecular speed when $T$ is increased to $700\ \mathrm{K}$?
Average molecular speed increases by a factor of $\sqrt{2}$ because it scales with $\sqrt{T}$.
Average molecular speed decreases because higher temperature increases collision frequency and slows molecules.
Average molecular speed is unchanged because molecular mass, not temperature, determines speed.
Average molecular speed doubles because it is directly proportional to absolute temperature.
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of temperature effects on molecular speed. Kinetic molecular theory relates average molecular speed to temperature through the relationship vrms = √(3RT/M), showing that root-mean-square speed is proportional to √T. In this scenario, doubling the temperature from 350 K to 700 K affects the average molecular speed of helium atoms. The correct choice, B, follows because when temperature doubles, the average molecular speed increases by a factor of √2 ≈ 1.414, as speed scales with the square root of absolute temperature. Choice A is incorrect because it assumes direct proportionality between speed and temperature, ignoring the square root relationship. In similar questions, remember that kinetic energy is proportional to T, but speed is proportional to √T due to the relationship KE = ½mv².
A researcher calibrates a pressure sensor using an ideal gas in a fixed-volume bulb. The bulb is held at $T=273\ \mathrm{K}$ and filled with $n=0.050\ \mathrm{mol}$ gas in a volume of $V=1.00\ \mathrm{L}$. Use $R=0.082\ \mathrm{L,atm,mol^{-1},K^{-1}}$.
Based on the ideal gas law, which pressure is most consistent with these conditions?
$P\approx 0.22\ \mathrm{atm}$
$P\approx 11.2\ \mathrm{atm}$
$P\approx 112\ \mathrm{atm}$
$P\approx 1.12\ \mathrm{atm}$
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of calculating pressure using the ideal gas law. The ideal gas law PV = nRT allows direct calculation of pressure when n, V, T, and R are known. In this scenario, substituting n = 0.050 mol, V = 1.00 L, T = 273 K, and R = 0.082 L·atm·mol⁻¹·K⁻¹ into P = nRT/V gives the pressure. The correct choice, A, follows because P = (0.050 × 0.082 × 273)/1.00 = 1.12 atm, matching the calculation precisely. Choice C is incorrect because it appears to have a decimal place error, giving a value 10 times too large. In similar questions, carefully track units and ensure all values are in consistent units (here, L for volume, atm for pressure via R's units).
In a diffusion assay at $298\ \mathrm{K}$, equal amounts (in moles) of two gases are released into identical long tubes under identical pressure and geometry. The time required for each gas to reach a detector at a fixed distance is recorded. Assume effusion/diffusion follows Graham’s law and that the measured time is inversely proportional to diffusion rate. Molar masses: $M(\mathrm{He})=4\ \mathrm{g/mol}$, $M(\mathrm{Ne})=20\ \mathrm{g/mol}$.
Which conclusion about the ratio of diffusion rates is most consistent with the model?
$\dfrac{r_{\mathrm{He}}}{r_{\mathrm{Ne}}}=\dfrac{M_{\mathrm{He}}}{M_{\mathrm{Ne}}}$, so helium diffuses 5 times faster.
$\dfrac{r_{\mathrm{He}}}{r_{\mathrm{Ne}}}=\sqrt{\dfrac{M_{\mathrm{Ne}}}{M_{\mathrm{He}}}}$, so helium diffuses about $\sqrt{5}$ times faster.
$\dfrac{r_{\mathrm{He}}}{r_{\mathrm{Ne}}}=\sqrt{\dfrac{M_{\mathrm{He}}}{M_{\mathrm{Ne}}}}$, so neon diffuses about $\sqrt{5}$ times faster.
$\dfrac{r_{\mathrm{He}}}{r_{\mathrm{Ne}}}=1$ because diffusion rate depends only on temperature, not molar mass.
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of Graham's law of diffusion. Graham's law states that the rate of diffusion is inversely proportional to the square root of molar mass: r ∝ 1/√M, or for two gases, r₁/r₂ = √(M₂/M₁). In this scenario, comparing helium (M = 4 g/mol) and neon (M = 20 g/mol) gives a molar mass ratio of 5. The correct choice, B, follows because rHe/rNe = √(MNe/MHe) = √(20/4) = √5 ≈ 2.24, meaning helium diffuses approximately √5 times faster than neon. Choice C is incorrect because it inverts the mass ratio under the square root, which would incorrectly predict neon diffusing faster. In similar questions, remember that lighter gases diffuse faster, and the rate ratio equals the square root of the inverse mass ratio.
A gas-phase reaction is monitored in a closed, rigid reactor at constant temperature. Initially, the reactor contains $1.0\ \mathrm{mol}$ of an ideal gas at $P_0$. A valve then opens to a second, identical evacuated rigid chamber, allowing the gas to expand freely until equilibrium. Temperature remains constant and no gas is lost.
Based on the ideal gas law applied to free expansion into a vacuum, what is the most consistent prediction for the final pressure in the two-chamber system?
The final pressure is $2P_0$ because the gas occupies twice the space and collides more often.
The final pressure is $\tfrac{1}{2}P_0$ because total volume doubles at constant $T$ and $n$.
The final pressure cannot be predicted without the gas molar mass.
The final pressure is $P_0$ because temperature and moles are unchanged.
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of free expansion into vacuum. When an ideal gas expands into vacuum at constant temperature, the process is isothermal with no work done, and the ideal gas law applies to the final equilibrium state. In this scenario, the gas initially in volume V expands to fill total volume 2V while maintaining constant T and n. The correct choice, C, follows because from PV = nRT, if volume doubles while T and n remain constant, pressure must halve to maintain the equality, giving Pfinal = P₀/2. Choice A is incorrect because it suggests pressure increases with volume, contradicting the inverse relationship at constant T and n. In similar questions involving free expansion, apply the ideal gas law to initial and final states, recognizing that doubling volume halves pressure when other variables are constant.
In a sealed, rigid 2.00 L reaction vessel (constant volume) containing an ideal gas, a researcher increases the temperature while maintaining the amount of gas constant. The pressure is recorded after thermal equilibration at each temperature. Use the ideal gas relationship at constant $V$ and $n$ (Gay-Lussac’s Law: $P \propto T$ in kelvin). Constants: $1\ \text{atm} = 101.3\ \text{kPa}$. Based on the scenario, what effect would increasing the temperature from 300 K to 450 K most likely have on the gas pressure?
Initial condition: $P_1 = 100\ \text{kPa}$ at $T_1=300\ \text{K}$.
Increase to $150\ \text{kPa}$ because $P_2 = P_1,(T_2/T_1)$ at constant volume
Decrease to $66.7\ \text{kPa}$ because pressure is inversely proportional to temperature at constant volume
Remain at $100\ \text{kPa}$ because pressure depends only on volume in a rigid vessel
Increase to $225\ \text{kPa}$ because pressure scales with $T^2$ for an ideal gas
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of Gay-Lussac's Law for a fixed volume system. Gay-Lussac's Law states that for an ideal gas at constant volume and amount, pressure is directly proportional to absolute temperature (P ∝ T or P₁/T₁ = P₂/T₂). In this scenario, the sealed rigid vessel maintains constant volume while temperature increases from 300 K to 450 K. The correct choice, B, follows because P₂ = P₁(T₂/T₁) = 100 kPa × (450 K/300 K) = 100 kPa × 1.5 = 150 kPa. Choice A is incorrect because it suggests inverse proportionality between pressure and temperature, which contradicts Gay-Lussac's Law. In similar questions, ensure to use absolute temperature (Kelvin) and verify that volume and amount of gas remain constant to apply this relationship correctly.
A closed container of fixed volume holds an ideal gas. The researcher increases the number of moles of gas by injecting additional gas while maintaining constant temperature (thermal bath at 298 K). Principle: at constant $T$ and $V$, the ideal gas law implies $P \propto n$. If the initial pressure is $P_1=1.0\ \text{atm}$ at $n_1=0.50\ \text{mol}$, what is the most consistent prediction for the pressure after increasing to $n_2=0.80\ \text{mol}$?
Constants: none needed.
$P_2=0.625\ \text{atm}$ because pressure is inversely proportional to moles at constant temperature
$P_2=1.6\ \text{atm}$ because $P_2=P_1,(n_2/n_1)$ at constant temperature and volume
$P_2=1.3\ \text{atm}$ because pressure increases by $0.3\ \text{atm}$ when moles increase by 0.30 mol
$P_2=2.56\ \text{atm}$ because pressure scales with $n^2$ for ideal gases at constant volume
Explanation
This question assesses understanding of gas laws and kinetic molecular theory (4B) in the context of pressure changes with varying amounts of gas. The ideal gas law shows that at constant temperature and volume, pressure is directly proportional to the number of moles (P ∝ n). In this scenario, moles increase from 0.50 mol to 0.80 mol while temperature (298 K) and volume remain constant. The correct choice, B, follows because P₂ = P₁(n₂/n₁) = 1.0 atm × (0.80 mol/0.50 mol) = 1.0 atm × 1.6 = 1.6 atm. Choice A is incorrect because it suggests inverse proportionality between pressure and moles, contradicting the ideal gas law. In similar questions, ensure temperature and volume are constant, then apply direct proportionality between pressure and amount of gas.