Consider a real gas described by the Van der Waals equation. Under which conditions would the term $n^2a/V^2$ (related to intermolecular attractions) be most significant relative to the term $nb$ (related to molecular volume), causing the real gas pressure to be significantly lower than the ideal gas pressure?

Low temperature and moderate pressure.

High temperature and high pressure.

Low temperature and low pressure.

Low temperature and moderate pressure.

High temperature and low pressure.

For a fixed amount of an ideal gas in a rigid container, if the absolute temperature is doubled, what happens to the average kinetic energy of the molecules, the root-mean-square (rms) speed, and the pressure exerted by the gas?

Average kinetic energy doubles, rms speed increases by a factor of $\sqrt{2}$, pressure doubles.

Average kinetic energy doubles, rms speed doubles, pressure doubles.

Average kinetic energy increases by a factor of $\sqrt{2}$, rms speed doubles, pressure increases by a factor of $\sqrt{2}$.

Average kinetic energy doubles, rms speed increases by a factor of $\sqrt{2}$, pressure increases by a factor of $\sqrt{2}$.

Gases — Hard Practice Quiz

A Chemistry cheat sheet for Gases — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Formula	Name	Characteristics
HCN	Hydrogen cyanide	Very toxic, slight odor of bitter almonds
H₂S	Hydrogen sulfide	Very toxic, odor of rotten eggs
CO	Carbon monoxide	Toxic, colorless, odorless
CO₂	Carbon dioxide	Colorless, odorless
CH₄	Methane	Colorless, odorless, flammable
C₂H₄	Ethene (Ethylene)	Colorless, ripens fruit
C₃H₈	Propane	Colorless, odorless, bottled gas
N₂O	Nitrous oxide	Colorless, sweet odor, laughing gas
NO₂	Nitrogen dioxide	Toxic, red-brown, irritating odor
NH₃	Ammonia	Colorless, pungent odor
SO₂	Sulfur dioxide	Colorless, irritating odor

$$\text{Table 10.1: Common Gases}$$

Units	Numerical Value
L-atm/mol-K	0.08206
J/mol-K (SI unit)	8.314
cal/mol-K	1.987
m³-Pa/mol-K (SI unit)	8.314
L-torr/mol-K	62.36

$$\text{Table 10.2: Gas Constant R}$$

Substance	a (L²-atm/mol²)	b (L/mol)
He	0.0341	0.02370
Ne	0.211	0.0171
Ar	1.34	0.0322
Kr	2.32	0.0398
Xe	4.19	0.0510
H₂	0.244	0.0266
N₂	1.39	0.0391
O₂	1.36	0.0318
Cl₂	6.49	0.0562
H₂O	5.46	0.0305
CH₄	2.25	0.0428
CO₂	3.59	0.0427
CCl₄	20.4	0.1383

$$\text{Table 10.3: Van der Waals Constants}$$

Pressure is the force acting on a given area.

$$P = \frac{F}{A}$$

Where: P = pressure, F = force, A = area

Boyle's Law states that for a fixed quantity of gas at constant temperature, volume is inversely proportional to pressure.

$$PV = \text{constant}$$

Where: P = pressure, V = volume

Charles's Law states that for a fixed quantity of gas at constant pressure, volume is directly proportional to absolute temperature.

$$\frac{V}{T} = \text{constant}$$

Where: V = volume, T = absolute temperature (K)

Avogadro's Law states that the volume of a gas at constant temperature and pressure is directly proportional to the number of moles.

$$V = \text{constant} \times n$$

Where: V = volume, n = number of moles

Ideal-Gas Equation.

$$PV = nRT$$

Where: P = pressure, V = volume, n = moles, R = gas constant, T = temperature (K)

The ideal-gas equation rearranged to show the gas constant R.

$$\frac{PV}{nT} = R$$

Where: R = gas constant

Boyle's Law relates initial and final states at constant n and T.

$$P_1V_1 = P_2V_2$$

Where: 1 = initial state, 2 = final state

Combined Gas Law for a fixed amount of gas (constant n).

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$

Where: 1 = initial state, 2 = final state

Moles per unit volume (concentration) derived from ideal-gas equation.

$$\frac{n}{V} = \frac{P}{RT}$$

Where: n/V = molar concentration

Density of a gas.

$$d = \frac{PM}{RT}$$

Where: d = density, M = molar mass

Calculating molar mass from gas density.

$$M = \frac{dRT}{P}$$

Where: M = molar mass, d = density

Dalton's Law of Partial Pressures.

$$P_{total} = P_1 + P_2 + P_3 + \dots$$

Where: P_{total} = total pressure, P_i = partial pressure of component i

Total pressure related to total moles.

$$P_{total} = (n_1 + n_2 + \dots)\frac{RT}{V} = n_t \frac{RT}{V}$$

Where: n_t = total moles

Partial pressure of a single component in a mixture.

$$P_1 = n_1 \frac{RT}{V}$$

Where: P_1 = partial pressure, n_1 = moles of component 1

Mole fraction of component 1.

$$X_1 = \frac{n_1}{n_t}$$

Where: X_1 = mole fraction

Partial pressure related to mole fraction and total pressure.

$$P_1 = X_1 P_{total}$$

Where: P_1 = partial pressure, X_1 = mole fraction, P_{total} = total pressure

Decomposition of potassium chlorate (example reaction).

$$2\text{KClO}_3(s) \rightarrow 2\text{KCl}(s) + 3\text{O}_2(g)$$

Collecting gas over water. Total pressure includes water vapor pressure.

$$P_{total} = P_{gas} + P_{H_2O}$$

Where: P_{H_2O} = vapor pressure of water

Pressure from kinetic-molecular theory.

$$P \propto \frac{n m (u_{rms})^2}{V}$$

Where: m = mass of molecule, u_{rms} = rms speed

Average kinetic energy of a molecule.

$$\epsilon = \frac{1}{2} m u^2$$

Where: \epsilon = kinetic energy, u = speed

Ideal-gas equation derived from kinetic-molecular theory.

$$PV = nRT$$

Root-mean-square (rms) speed of gas molecules.

$$u_{rms} = \sqrt{\frac{3RT}{M}}$$

Where: M = molar mass (kg/mol for SI units)

Most probable speed of gas molecules.

$$u_{mp} = \sqrt{\frac{2RT}{M}}$$

Where: u_{mp} = most probable speed

Graham's Law of Effusion.

$$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$$

Where: r = rate of effusion, M = molar mass

Ratio of effusion rates equals ratio of rms speeds.

$$\frac{r_1}{r_2} = \frac{u_{rms1}}{u_{rms2}}$$

Where: r = effusion rate, u_{rms} = rms speed

Compressibility factor for 1 mole of an ideal gas.

$$\frac{PV}{RT} = 1$$

Where: Z = compressibility factor (1 for ideal gas)

Van der Waals equation for real gases.

$$\left(P + \frac{n^2a}{V^2}\right)(V - nb) = nRT$$

Where: a, b = van der Waals constants

Practice quiz

A $0.500 \text{ g}$ sample of an unknown gas is collected over water at $25 \text{ \textdegree C}$ in a $2.00 \text{ L}$ flask. The total pressure in the flask is $603 \text{ torr}$. If the vapor pressure of water at $25 \text{ \textdegree C}$ is $23.8 \text{ torr}$, identify the gas from Table 10.1.
- Methane ($CH_4$)
- Carbon dioxide ($CO_2$)
- Nitrous oxide ($N_2O$)
- Hydrogen sulfide ($H_2S$)
Answer: Methane ($CH_4$)
Calculate the pressure exerted by $1.00 \text{ mol}$ of $CO_2$ gas in a $0.500 \text{ L}$ container at $273 \text{ K}$ using both the ideal gas law and the Van der Waals equation. What is the percentage difference between the two calculated pressures? Use $a = 3.59 \text{ L}^2 \cdot \text{atm/mol}^2$ and $b = 0.0427 \text{ L/mol}$ for $CO_2$.
- $10.5\%$
- $15.2\%$
- $22.7\%$
- $30.1\%$
Answer: $22.7\%$
At $0 \text{ \textdegree C}$ and $1.00 \text{ atm}$, an unknown gas A has a density of $1.96 \text{ g/L}$. If this gas A effuses through a pinhole at a rate that is $1.50$ times faster than another unknown gas B, what is the molar mass of gas B?
- $29.3 \text{ g/mol}$
- $43.9 \text{ g/mol}$
- $65.9 \text{ g/mol}$
- $98.8 \text{ g/mol}$
Answer: $98.8 \text{ g/mol}$
A $10.0 \text{ L}$ container at $298 \text{ K}$ holds a mixture of $5.00 \text{ g}$ of $N_2$ and an unknown mass of $O_2$. The total pressure in the container is $2.50 \text{ atm}$. What is the partial pressure of $O_2$?
- $0.436 \text{ atm}$
- $1.25 \text{ atm}$
- $2.06 \text{ atm}$
- $2.50 \text{ atm}$
Answer: $2.06 \text{ atm}$
At what temperature (in Kelvin) would $H_2$ gas have the same root-mean-square (rms) speed as $O_2$ gas at $300 \text{ K}$?
- $18.9 \text{ K}$
- $300 \text{ K}$
- $1500 \text{ K}$
- $4800 \text{ K}$
Answer: $18.9 \text{ K}$
Consider a real gas described by the Van der Waals equation. Under which conditions would the term $n^2a/V^2$ (related to intermolecular attractions) be most significant relative to the term $nb$ (related to molecular volume), causing the real gas pressure to be significantly lower than the ideal gas pressure?
- High temperature and high pressure.
- Low temperature and low pressure.
- Low temperature and moderate pressure.
- High temperature and low pressure.
Answer: Low temperature and moderate pressure.
A fixed amount of an ideal gas initially occupies a volume $V_1$ at pressure $P_1$ and temperature $T_1$, with a density $d_1$. If the pressure is doubled ($P_2 = 2P_1$) and the absolute temperature is halved ($T_2 = T_1/2$), what is the new density $d_2$ in terms of $d_1$?
- $d_2 = d_1/4$
- $d_2 = d_1/2$
- $d_2 = 2d_1$
- $d_2 = 4d_1$
Answer: $d_2 = 4d_1$
A $10.0 \text{ g}$ sample of solid potassium chlorate ($KClO_3$) is completely decomposed in a $5.00 \text{ L}$ container at $200 \text{ \textdegree C}$. The reaction is $2KClO_3(s) \rightarrow 2KCl(s) + 3O_2(g)$. What is the total pressure of oxygen gas produced in the container?
- $0.475 \text{ atm}$
- $0.950 \text{ atm}$
- $1.42 \text{ atm}$
- $1.90 \text{ atm}$
Answer: $0.950 \text{ atm}$
It takes $45.0 \text{ s}$ for $1.00 \text{ mol}$ of $CH_4$ gas to effuse through a porous barrier. How long would it take for $1.00 \text{ mol}$ of $SO_2$ gas to effuse through the same barrier under identical conditions?
- $22.5 \text{ s}$
- $45.0 \text{ s}$
- $89.9 \text{ s}$
- $180 \text{ s}$
Answer: $89.9 \text{ s}$
For a fixed amount of an ideal gas in a rigid container, if the absolute temperature is doubled, what happens to the average kinetic energy of the molecules, the root-mean-square (rms) speed, and the pressure exerted by the gas?
- Average kinetic energy doubles, rms speed increases by a factor of $\sqrt{2}$, pressure doubles.
- Average kinetic energy doubles, rms speed doubles, pressure doubles.
- Average kinetic energy increases by a factor of $\sqrt{2}$, rms speed doubles, pressure increases by a factor of $\sqrt{2}$.
- Average kinetic energy doubles, rms speed increases by a factor of $\sqrt{2}$, pressure increases by a factor of $\sqrt{2}$.
Answer: Average kinetic energy doubles, rms speed increases by a factor of $\sqrt{2}$, pressure doubles.

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Gases — Hard Practice Quiz

Formulas & key concepts

Pressure is the force acting on a given area.

Ideal-Gas Equation.

Combined Gas Law for a fixed amount of gas (constant n).

Density of a gas.

Calculating molar mass from gas density.

Dalton's Law of Partial Pressures.

Total pressure related to total moles.

Partial pressure of a single component in a mixture.

Mole fraction of component 1.

Decomposition of potassium chlorate (example reaction).

Pressure from kinetic-molecular theory.

Average kinetic energy of a molecule.

Ideal-gas equation derived from kinetic-molecular theory.

Root-mean-square (rms) speed of gas molecules.

Most probable speed of gas molecules.

Graham's Law of Effusion.

Ratio of effusion rates equals ratio of rms speeds.

Compressibility factor for 1 mole of an ideal gas.

Van der Waals equation for real gases.

Practice quiz

Select a subject

Subjects

Print

Gases — Hard Practice Quiz

Formulas & key concepts

Pressure is the force acting on a given area.

Ideal-Gas Equation.

Combined Gas Law for a fixed amount of gas (constant n).

Density of a gas.

Calculating molar mass from gas density.

Dalton's Law of Partial Pressures.

Total pressure related to total moles.

Partial pressure of a single component in a mixture.

Mole fraction of component 1.

Decomposition of potassium chlorate (example reaction).

Pressure from kinetic-molecular theory.

Average kinetic energy of a molecule.

Ideal-gas equation derived from kinetic-molecular theory.

Root-mean-square (rms) speed of gas molecules.

Most probable speed of gas molecules.

Graham's Law of Effusion.

Ratio of effusion rates equals ratio of rms speeds.

Compressibility factor for 1 mole of an ideal gas.

Van der Waals equation for real gases.

Practice quiz

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