General Chemistry Study Notes: Gases, Gas Laws, and Related Calculations

Gases and Gas Laws

Introduction to Gas Laws

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container. The behavior of gases is described by several empirical laws, which relate pressure, volume, temperature, and amount of gas. These laws are essential for understanding chemical reactions involving gases and for performing calculations in laboratory and real-world settings.

• Standard Temperature and Pressure (STP): STP is defined as 1 atm pressure and 0°C (273 K).

• Unit Conversions: 1 atm = 760 mm Hg = 760 torr

Key Gas Laws and Equations

• Ideal Gas Law: Relates pressure, volume, temperature, and moles of a gas. $PV = nRT$ Where:

  • P = pressure (atm)

  • V = volume (L)

  • n = moles of gas

  • R = ideal gas constant (0.08206 L·atm·mol-1·K-1)

  • T = temperature (K)

• Boyle's Law: At constant temperature, pressure and volume are inversely related. $P_1V_1 = P_2V_2$

• Charles's Law: At constant pressure, volume and temperature are directly related. $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro's Law: At constant temperature and pressure, volume and moles are directly related. $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Dalton's Law of Partial Pressures: The total pressure of a mixture of gases is the sum of the partial pressures of each component. $P_{\text{total}} = P_1 + P_2 + \ldots$ Partial pressure of a component: $P_i = X_i P_{\text{total}}$ Where $X_i$ is the mole fraction of component $i$.

• Density of a Gas: $d = \frac{PM}{RT}$ Where $M$ is the molar mass.

• Graham's Law of Effusion: The rate of effusion of a gas is inversely proportional to the square root of its molar mass. $\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$

Applications and Example Problems

Volume and Pressure Changes (Boyle's Law)

Boyle's Law is used to calculate the change in volume or pressure of a gas when the other variable changes, provided temperature and amount of gas remain constant.

• Example: If the initial volume of a gas cylinder is 750.0 mL and the pressure changes from 840.0 mm Hg to 360 mm Hg, the final volume can be calculated using $P_1V_1 = P_2V_2$.

Volume and Temperature Changes (Charles's Law)

Charles's Law allows calculation of the new volume or temperature of a gas when the other variable changes at constant pressure.

• Example: A 5.00 L balloon of gas at 25°C is cooled to 0°C. The new volume is found using $\frac{V_1}{T_1} = \frac{V_2}{T_2}$, with temperatures converted to Kelvin.

Combined Gas Law

When pressure, volume, and temperature all change, the combined gas law is used:

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

• Example: If a gas is compressed to a new volume and pressure, the new temperature can be calculated.

Calculating Moles of Gas (Ideal Gas Law)

The ideal gas law can be rearranged to solve for the number of moles:

• $n = \frac{PV}{RT}$

• Example: How many moles of gas are added to a balloon if the volume changes and the initial moles are known?

Partial Pressures and Gas Mixtures (Dalton's Law)

Dalton's Law is used to determine the pressure contributed by each gas in a mixture.

• Example: If a mixture contains 78% nitrogen at a certain pressure and 22% carbon dioxide at another pressure, the total pressure is the sum of the partial pressures.

• Example: In a cylinder with radon, nitrogen, and helium, the pressure due to helium is found by subtracting the pressures of the other gases from the total pressure.

Collecting Gases Over Water

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

• $P_{\text{gas}} = P_{\text{total}} - P_{\text{water vapor}}$

• Example: If the vapor pressure of water at 25°C is 23.8 mm Hg and the barometric pressure is 742.5 mm Hg, the pressure of dry hydrogen gas is $742.5 - 23.8 = 718.7$ mm Hg.

Stoichiometry of Gaseous Reactions

Gas volumes at STP can be related directly to moles using the molar volume (22.4 L/mol at STP). Balanced chemical equations allow calculation of volumes of reactants and products.

• Example: For the decomposition of ammonia: $2NH_3(g) \rightarrow N_2(g) + 3H_2(g)$ If 15.0 L of nitrogen is formed at STP, 45.0 L of hydrogen will be produced (using stoichiometric ratios).

• Example: For the formation of water: $2H_2(g) + O_2(g) \rightarrow 2H_2O(g)$ 8.0 L of hydrogen requires 4.0 L of oxygen at STP.

Gas Density and Molar Mass

The density of a gas can be calculated using the ideal gas law:

• $d = \frac{PM}{RT}$

• Example: Calculate the density of SF6 at 27°C and 0.500 atm.

Effusion and Graham's Law

Graham's Law relates the rate of effusion of two gases to their molar masses.

• $\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$

• Example: If a compound effuses 0.411 times as fast as neon, its molar mass can be calculated and the molecular formula deduced.

Determining Molecular Formula from Gas Data

Experimental data (mass, volume, temperature, pressure) can be used to determine the molar mass and thus the molecular formula of a compound.

• Example: A 1.325 g sample of vapor occupies 368 mL at 114°C and 946 mm Hg. The empirical formula is NO2. Use the ideal gas law to find the molar mass and deduce the molecular formula.

Summary Table: Key Gas Laws and Relationships

Law/Equation	Relationship	Formula	Variables
Boyle's Law	P and V (T constant)	$P_1V_1 = P_2V_2$	Pressure, Volume
Charles's Law	V and T (P constant)	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Volume, Temperature
Avogadro's Law	V and n (P, T constant)	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	Volume, Moles
Ideal Gas Law	P, V, n, T	$PV = nRT$	Pressure, Volume, Moles, Temperature
Dalton's Law	Partial Pressures	$P_{\text{total}} = P_1 + P_2 + \ldots$	Pressure
Graham's Law	Effusion Rate	$\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$	Molar Mass
Gas Density	Density	$d = \frac{PM}{RT}$	Pressure, Molar Mass, Temperature

Additional info:

• All calculations involving temperature must use Kelvin: $K = ^\circ C + 273.15$

• When using the ideal gas law, ensure units are consistent (L, atm, K, mol).

• For stoichiometry at STP, 1 mol of any ideal gas occupies 22.4 L.