Properties and Laws of Gases

Combined Gas Law

The combined gas law relates the pressure, volume, and temperature of a fixed amount of gas. It is useful for calculations where the amount of gas remains constant but other conditions change.

• Formula:

• Key Point: Temperatures must be in Kelvin for calculations.

• Example: If a gas at 18°C (291 K), 43.0 L, and 765 torr changes to 26.0 L and 86°C (359 K), the new pressure can be found using the combined gas law.

Mole Fraction

The mole fraction is the ratio of the number of moles of a component to the total number of moles in a mixture.

• Formula:

• Key Point: The sum of all mole fractions in a mixture equals 1.

• Example: If and , then .

Ideal Gas Law and Applications

The ideal gas law relates pressure, volume, temperature, and number of moles of a gas.

• Formula:

• Key Point: R = 0.0821 L·atm/(mol·K) or 62.36 L·torr/(mol·K)

• Application: Used to calculate the number of atoms/molecules in a sample, density, or molar mass.

• Example: Calculating the number of argon atoms in a fluorescent tube using total pressure and partial pressure assumptions.

Dumas Method for Molar Mass

The Dumas method determines the molar mass of a volatile liquid by vaporizing it and measuring mass, volume, pressure, and temperature.

• Formula:

• Key Point: All measurements must be in SI units.

• Example: Given mass of vapor, flask volume, pressure, and temperature, calculate molar mass.

Gas Density

Density of a gas can be calculated using the ideal gas law.

• Formula:

• Key Point: M is the molar mass of the gas.

• Example: Find the density of phosgene gas at given pressure and temperature.

Gas Collection Over Water

When collecting gas over water, the total pressure includes both the gas and water vapor.

• Formula:

• Key Point: Use the corrected pressure to find moles using the ideal gas law.

• Example: Calculate moles of butane collected over water.

Effusion and Graham's Law

Effusion is the process by which gas molecules escape through a small hole. Graham's law relates the rate of effusion to molar mass.

• Formula:

• Key Point: Lighter gases effuse faster than heavier gases.

• Example: Use effusion times to calculate the molar mass of an unknown gas.

Root-Mean-Square Speed ()

The root-mean-square speed is a measure of the average speed of gas molecules.

• Formula:

• Key Point: Increasing temperature increases ; increasing molar mass decreases $v_{rms}$.

• Example: Effects of pressure, volume, and temperature changes on .

Kinetic Molecular Theory and Gas Comparisons

Kinetic molecular theory explains the behavior of gases in terms of particle motion and energy.

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

• Comparison Table:

Thermodynamics and Calorimetry

First Law of Thermodynamics

The first law states that energy cannot be created or destroyed, only transformed.

• Formula:

• Key Point: Internal energy change is the sum of heat and work.

• Example: Energy is conserved in all physical and chemical processes.

Internal Energy vs. Enthalpy

Internal energy () and enthalpy () are related but not always equal. At constant pressure, $\Delta H$ includes energy for expansion against atmospheric pressure.

• Formula:

• Key Point: For reactions with no volume change, .

Endothermic and Exothermic Reactions

Endothermic reactions absorb heat, making surroundings feel colder. Exothermic reactions release heat, warming the surroundings.

• Key Point: Endothermic: ; Exothermic:

• Example: Dissolving ammonium nitrate in water is endothermic.

Specific Heat and Calorimetry

Specific heat is the amount of heat required to raise the temperature of 1 gram of a substance by 1°C.

• Formula:

• Key Point: Used to calculate temperature changes and heat absorbed/released.

• Example: Calculating final temperature after heat absorption.

Bomb Calorimetry

Bomb calorimeters measure the heat of combustion at constant volume.

• Formula:

• Key Point: Used to determine enthalpy of combustion.

• Example: Burning octane and measuring temperature rise.

Potential Energy Calculations

Potential energy due to gravity is calculated for objects at a height above a reference point.

• Formula:

• Key Point: m = mass (kg), g = acceleration due to gravity (9.8 m/s2), h = height (m)

• Example: Calculating the energy of a diver at the apex of a dive.

Advanced Gas Behavior

Real Gases and van der Waals Equation

Real gases deviate from ideal behavior at high pressures and low temperatures. The van der Waals equation corrects for intermolecular forces and molecular volume.

• Formula:

• Key Point: a = measure of attraction; b = volume correction.

• Example: Calculating pressure for NH3 using both ideal and real gas equations.

Enthalpy and Heat of Reaction

Standard Enthalpy of Formation

The standard enthalpy of formation () is the enthalpy change when one mole of a compound forms from its elements in their standard states.

• Formula:

• Key Point: Used to calculate heat released or absorbed in reactions.

• Example: Calculating total heat emitted during combustion of a fuel mixture.

Tables and Data Interpretation

Sample Table: Gas Properties Comparison

Sample Table: Enthalpy of Formation

Key Equations Summary

• Combined Gas Law:

• Ideal Gas Law:

• Density of Gas:

• Root-Mean-Square Speed:

• Graham's Law:

• Specific Heat:

• Potential Energy:

• van der Waals Equation:

• Enthalpy of Reaction:

Additional info:

• Some problems involve practical applications such as calorimetry, energy conversions, and real gas behavior.

• Graphical data (velocity distributions) illustrate molecular speed differences and effusion rates.

• Tables are used for comparison of gas properties and enthalpy values.