General Chemistry: Gases, Chemical Reactions, and Stoichiometry Study Notes

Gases and Gas Laws

Properties of Gases

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container, low density, and high compressibility. The behavior of gases can be described using several empirical laws and the kinetic molecular theory.

• Compressibility: Gases can be compressed much more easily than solids or liquids due to the large distances between particles.

• Expansion: Gases expand to fill the volume of their container.

• Low Density: The density of gases is much lower than that of solids and liquids.

Gas Laws

Gas laws describe the relationships between pressure, volume, temperature, and amount of gas.

• Boyle's Law: At constant temperature, the pressure and volume of a gas are inversely proportional.

• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature (in Kelvin).

• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas.

• Ideal Gas Law: Combines the above laws into one equation:

$PV = nRT$

• P: Pressure (atm)

• V: Volume (L)

• n: Number of moles

• R: Ideal gas constant (0.0821 L·atm·mol-1·K-1)

• T: Temperature (K)

Dalton's Law of Partial Pressures

In a mixture of non-reacting gases, the total pressure is the sum of the partial pressures of each individual gas.

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

• Partial Pressure: The pressure exerted by a single component in a mixture of gases.

Collecting Gases Over Water

When collecting gases over water, the total pressure is the sum of the gas pressure and the vapor pressure of water.

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

Kinetic Molecular Theory and Molecular Speeds

The kinetic molecular theory explains the behavior of gases in terms of the motion of their particles.

• Root Mean Square Speed (urms): The average speed of gas molecules, related to temperature and molar mass:

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

• R: Gas constant (8.314 J·mol-1·K-1)

• T: Temperature (K)

• M: Molar mass (kg/mol)

• Diffusion: The mixing of gases due to molecular motion.

• Effusion: The escape of gas molecules through a small hole.

Graham's Law of Effusion

Graham's law relates the rates of effusion of two gases to their molar masses:

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

• Gases with lower molar mass effuse and diffuse faster.

Real Gases

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

• Van der Waals Equation: Adjusts the ideal gas law for real gases:

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

• a, b: Empirical constants for each gas.

Chemical Reactions and Stoichiometry

Types of Chemical Reactions

Chemical reactions involve the transformation of reactants into products. Common types include combination, decomposition, single displacement, double displacement, and combustion reactions.

• Combination: Two or more substances combine to form one product.

• Decomposition: A single compound breaks down into two or more simpler substances.

• Single Displacement: One element replaces another in a compound.

• Double Displacement: Exchange of ions between two compounds.

• Combustion: A substance reacts with oxygen, releasing energy.

Balancing Chemical Equations

Balancing ensures the same number of each type of atom on both sides of the equation, in accordance with the law of conservation of mass.

• Adjust coefficients to balance atoms.

• Never change subscripts in chemical formulas.

Stoichiometry

Stoichiometry involves calculations based on the relationships between reactants and products in a balanced chemical equation.

• Mole Ratio: Derived from the coefficients of a balanced equation.

• Limiting Reactant: The reactant that is completely consumed first, limiting the amount of product formed.

• Theoretical Yield: The maximum amount of product that can be formed from the limiting reactant.

• Percent Yield: The ratio of actual yield to theoretical yield, expressed as a percentage.

$\text{Percent Yield} = \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \times 100\%$

Example: Reaction of Ca(HSO4)2 with H2O

Consider the reaction:

$\mathrm{Ca(HSO_4)_2 (aq) + 2 H_2O (l) \rightarrow Ca^{2+} (aq) + 2 HSO_4^- (aq) + 2 H_2O (l)}$

• This is a dissociation reaction in aqueous solution.

• Classified as a strong electrolyte because it dissociates completely in water.

Laboratory Calculations and Applications

Gas Collection and Stoichiometry

When collecting gases in the laboratory, it is important to account for the total pressure, including water vapor if collected over water. Use stoichiometry to relate the amount of reactant to the volume of gas produced.

• Use the ideal gas law to calculate the volume or pressure of gases collected.

• Adjust for water vapor pressure if necessary.

Sample Calculation: Partial Pressure

Given a mixture of gases, calculate the partial pressure of each component using mole fractions:

$P_i = X_i P_{total}$

• Xi: Mole fraction of component i

• Ptotal: Total pressure

Sample Calculation: Root Mean Square Speed

Calculate the root mean square speed of a gas molecule at a given temperature:

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

• For example, for H2 at 298 K, use the molar mass in kg/mol.

Summary Table: Gas Law Relationships

Law	Equation	Variables Held Constant	Relationship
Boyle's Law	$P_1V_1 = P_2V_2$	n, T	$P \propto \frac{1}{V}$
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	n, P	$V \propto T$
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	P, T	$V \propto n$
Ideal Gas Law	$PV = nRT$	—	All variables related

Key Concepts and True/False Review

• All of the H2 gas molecules in a 1.0 L container at 25°C move faster than all of the N2 gas molecules at 1.0 L, 25°C. False (There is a distribution of speeds.)

• At the same temperature, lighter gases have higher average molecular speeds. True

• Replacing half the gas in a container with another gas at the same temperature and pressure does not change the total pressure. True

• The pressure of a gas mixture is the sum of the partial pressures of each component. True

Laboratory Application: Acid-Base Stoichiometry

Example: Reaction of HCl and CaCO3

To determine the amount of CaCO3 in a sample, react it with excess HCl and measure the amount of CO2 produced.

$\mathrm{CaCO_3 (s) + 2 HCl (aq) \rightarrow CaCl_2 (aq) + CO_2 (g) + H_2O (l)}$

• Calculate moles of HCl used and relate to moles of CaCO3 via stoichiometry.

• Percent CaCO3 can be determined from the mass of the original sample.

Additional info:

• Some calculations and answers were inferred from handwritten solutions and standard chemistry procedures.

• All equations are provided in standard LaTeX format for clarity.