chapter 9 lec 2

Gases: Properties and Laws

Dalton’s Law of Partial Pressures

When dealing with mixtures of gases, such as air, the total pressure exerted by the mixture is the sum of the partial pressures of each individual gas. Each gas behaves as if it occupies the container alone.

• Partial Pressure: The pressure exerted by a single gas in a mixture, calculated as if it were the only gas present.

• Dalton’s Law: $P_T = P_A + P_B + P_C + \ldots$

• Partial Pressure Formula: $P_A = \frac{n_A R T}{V}$, $P_B = \frac{n_B R T}{V}$

• Total Pressure: $P_T = (n_A + n_B + n_C + \ldots) \frac{R T}{V} = n_{\text{total}} \frac{R T}{V}$

Mole Fraction (X): The ratio of moles of a component to the total moles in the mixture.

• $X_A = \frac{n_A}{n_{\text{total}}}$

• $P_A = X_A P_T$

• The sum of all mole fractions in a mixture equals 1: $X_A + X_B + X_C + \ldots = 1$

Example: Calculating Mass of Argon in a Gas Mixture

• Total pressure: 663 mmHg; Partial pressures: He = 341 mmHg, Ne = 112 mmHg

• Find $P_{Ar}$: $P_{Ar} = 663 - 341 - 112 = 210$ mmHg

• Convert to atm: $P_{Ar} = 210 \times \frac{1}{760} = 0.2763$ atm

• Use ideal gas law: $n = \frac{PV}{RT} = \frac{0.2763 \times 1.00}{0.0821 \times 298} = 0.01129$ mol

• Mass: $0.01129 \text{ mol} \times 39.94 \text{ g/mol} = 0.451$ g Ar

Example: Heliox Mixture in a Scuba Tank

• Given: 24.2 g He, 4.32 g O2, 12.5 L, 298 K

• Convert to moles: He = 6.045 mol, O2 = 0.135 mol

• Mole fractions: $X_{He} = 0.978$, $X_{O_2} = 0.022$

• Total pressure: $P_T = \frac{(6.18)(0.0821)(298)}{12.5} = 12.1$ atm

• Partial pressures: $P_{He} = 11.8$ atm, $P_{O_2} = 0.26$ atm

Collecting Gases Over Water

When a gas is collected over water, the total pressure includes both the gas and water vapor. The partial pressure of the collected gas is found by subtracting the vapor pressure of water from the total pressure.

• Formula: $P_{\text{total}} = P_{\text{gas}} + P_{\text{H}_2O}$

• Look up $P_{\text{H}_2O}$ at the collection temperature.

Example: Hydrogen Collected Over Water

• Total pressure: 758.2 mmHg at 25°C; $P_{\text{H}_2O}$ at 25°C = 23.78 mmHg

• $P_{H_2} = 758.2 - 23.78 = 734.4$ mmHg

• Convert to atm and use $n = \frac{PV}{RT}$ to find moles and mass of H2

Effusion and Diffusion of Gases

Diffusion

Diffusion is the gradual mixing of molecules of one gas with another due to their kinetic energy. Lighter gases diffuse faster than heavier ones.

• Example: Ammonia and HCl gases react to form a visible white ring of NH4Cl closer to the HCl source, since NH3 diffuses faster.

Graham’s Law of Diffusion and Effusion

Thomas Graham found that the rate of diffusion (or effusion) of a gas is inversely proportional to the square root of its molar mass.

• Graham’s Law: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

• $r_1, r_2$ = rates of diffusion/effusion; $M_1, M_2$ = molar masses

Example: Unknown Gas Diffusion Rate

• Unknown gas diffuses at 0.462 times the rate of N2

• $M_{\text{unknown}} = \frac{28.01}{(0.462)^2} = 131$ g/mol

Example: H2 vs. Kr Diffusion

• $\frac{r_{Kr}}{r_{H_2}} = \sqrt{\frac{2.016}{83.80}} = 0.155$

• H2 diffuses 6.45 times faster than Kr

Effusion is the process by which a gas escapes through a small hole. The same law applies as for diffusion.

• He balloons deflate faster than air-filled ones due to higher effusion rate of He.

Kinetic-Molecular Theory of Gases

Postulates of the Kinetic-Molecular Theory

This theory explains the macroscopic properties of gases in terms of the motion of their molecules.

• Gas molecules are far apart compared to their size; their volume is negligible.

• They move in constant, random, straight-line motion and collide elastically.

• No attractive or repulsive forces exist between molecules.

• Average kinetic energy is proportional to absolute temperature (K).

Average Kinetic Energy:

• $KE_{\text{ave}} = \frac{1}{2} m \overline{u^2}$

• $\overline{u^2}$ is the mean square speed: $\overline{u^2} = \frac{u_1^2 + u_2^2 + \ldots + u_N^2}{N}$

• At the same temperature, all gases have the same average kinetic energy.

Applications to Gas Laws

• Compressibility: Gases can be compressed due to large intermolecular spaces.

• Boyle’s Law: Pressure is inversely proportional to volume due to collision frequency.

• Charles’s Law: Increasing temperature increases kinetic energy, causing expansion.

• Avogadro’s Law: Equal volumes of gases at the same T and P contain equal numbers of molecules.

• Dalton’s Law: Each gas exerts pressure independently if no intermolecular forces exist.

Molecular Speeds and Root-Mean-Square Speed

Distribution of Molecular Speeds

At a given temperature, molecules have a range of speeds. The most probable speed increases with temperature, and lighter molecules move faster than heavier ones.

• He (4.003 g/mol): ~1100 m/s

• N2 (28.02 g/mol): ~490 m/s

• Cl2 (70.90 g/mol): ~250 m/s

Root-Mean-Square (rms) Speed

The rms speed is a measure of the average speed of gas molecules at a given temperature.

• Formula: $u_{\text{rms}} = \sqrt{\frac{3RT}{MW}}$

• R = 8.314 J/(mol·K); MW in kg/mol; $u_{\text{rms}}$ in m/s

Example: Oxygen at 25°C

• $u_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 298}{0.032}} = 482$ m/s

• Converted to mph: 1084 mi/hr

Non-Ideal Gas Behavior

Deviation from Ideal Gas Law

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of molecules.

• At low pressure, gases behave ideally: $PV/RT = 1$

• At high pressure, attractive and repulsive forces become significant, causing deviations.

van der Waals Equation

To account for non-ideal behavior, the van der Waals equation introduces correction factors for pressure and volume:

• Equation: $[P + a \frac{n^2}{V^2}](V - nb) = nRT$

• a = measure of intermolecular attraction; b = measure of finite molecular volume

Gas	a (atm·L2/mol2)	b (L/mol)
He	0.034	0.0237
N2	1.39	0.0391
O2	1.36	0.0318
H2O	5.46	0.0305

Example: Oxygen at High Pressure

• 200 mol O2, 58.29 L, 355.0 K

• Ideal gas: $P = \frac{nRT}{V} = 100.0$ atm

• van der Waals: $P = 96.2$ atm (lower due to intermolecular attractions)

Conclusion: Real gases exert less pressure than predicted by the ideal gas law at high concentrations due to attractive forces.