General Chemistry Study Guide: Gases, Thermochemistry, and Quantum-Mechanical Model of the Atom

Chapter 6. Gases

6.5 Applications of the Ideal Gas Law: Molar Volume, Density, and Molar Mass of a Gas

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and amount of gas. It is widely used to determine the properties of gases under various conditions.

• Ideal Gas Law: $PV = nRT$

• Molar Volume: The volume occupied by one mole of an ideal gas at standard temperature and pressure (STP) is 22.4 L.

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

• Molar Mass from Gas Data: $M = \frac{dRT}{P}$

• Relationship: The ideal gas law can be rearranged to solve for any variable, allowing calculation of molar mass, density, or volume.

• Example: Calculate the density of oxygen gas at 1.00 atm and 0°C.

6.6 Mixtures of Gases and Partial Pressures

When gases are mixed, each gas exerts a pressure independently of the others. The total pressure is the sum of the partial pressures of each component.

• Partial Pressure: The pressure exerted by a single gas in a mixture.

• Dalton's Law of Partial Pressures: $P_{total} = P_1 + P_2 + P_3 + \ldots$

• Mole Fraction: $X_i = \frac{n_i}{n_{total}}$; $P_i = X_i P_{total}$

• Application: Used to calculate the pressure of a gas collected over water or in air mixtures.

• Example: Find the partial pressure of nitrogen in air if air is 78% nitrogen by mole.

6.7 Gases in Chemical Reactions: Stoichiometry Revisited

Stoichiometry involving gases often uses the ideal gas law to relate moles of gas to volume under given conditions.

• Gas Stoichiometry: Use $PV = nRT$ to convert between volume and moles in reactions involving gases.

• Application: Calculating the volume of gas produced or consumed in a chemical reaction.

• Example: What volume of CO2 is produced at STP from the combustion of 1 mol of methane?

6.8 Kinetic Molecular Theory: A Model for Gases

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

• Postulates: Gases consist of tiny particles in constant, random motion; collisions are elastic; volume of particles is negligible; no intermolecular forces.

• Explains: Boyle's, Charles's, Avogadro's, and Dalton's laws.

• Average Kinetic Energy: $KE_{avg} = \frac{3}{2}RT$

• Root Mean Square Velocity: $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Distribution of Speeds: At a given temperature, not all molecules move at the same speed; speeds follow a distribution.

• Example: Compare the average speed of H2 and O2 molecules at the same temperature.

6.9 Effusion and Diffusion of Gases

Effusion is the escape of gas through a small hole; diffusion is the mixing of gases. Both are explained by kinetic molecular theory.

• Graham's Law of Effusion: $\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

• Mean Free Path: The average distance a molecule travels between collisions.

• Application: Used to separate isotopes or gases of different molar masses.

• Example: Calculate the relative rate of effusion of helium and argon.

6.10 Real Gases: The Effects of Size and Intermolecular Forces

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and finite molecular size.

• Van der Waals Equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

• Parameters: 'a' corrects for intermolecular attractions; 'b' corrects for finite molecular volume.

• Application: Used to predict the behavior of real gases under non-ideal conditions.

• Example: Compare the pressure of a real gas and an ideal gas under the same conditions.

Equations and Relationships in Chapter 6

• Ideal Gas Law: $PV = nRT$

• Dalton's Law: $P_{total} = P_1 + P_2 + \ldots$

• Mole Fractions: $X_i = \frac{n_i}{n_{total}}$; $P_i = X_i P_{total}$

• Average Kinetic Energy: $KE_{avg} = \frac{3}{2}RT$

• Root Mean Square Velocity: $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Graham's Law: $\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

• Van der Waals Equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

Chapter 7. Thermochemistry

7.2 The Nature of Energy: Key Definitions

Thermochemistry studies the energy changes that accompany chemical reactions and physical changes.

• Energy: The capacity to do work or transfer heat.

• Kinetic Energy: Energy due to motion; $KE = \frac{1}{2}mv^2$

• Potenial Energy: Energy due to position or composition.

• Thermal Energy: Energy associated with temperature.

• System and Surroundings: The part of the universe under study and everything else, respectively.

• Units: Joule (J), calorie (cal), kilojoule (kJ), kilocalorie (kcal).

7.3 The First Law of Thermodynamics: There Is No Free Lunch

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

• First Law: $\Delta E = q + w$

• Internal Energy (E): The total energy of a system.

• Work (w): Energy transfer due to force acting over a distance; $w = -P\Delta V$

• Heat (q): Energy transfer due to temperature difference.

• Sign Conventions: $q$ and $w$ are positive if energy flows into the system.

7.4 Quantifying Heat and Work

Heat capacity and specific heat are used to quantify the amount of heat required to change the temperature of a substance.

• Heat Capacity (C): Amount of heat required to raise the temperature of an object by 1°C.

• Specific Heat (c): Amount of heat required to raise 1 g of a substance by 1°C.

• Formula: $q = m \times c \times \Delta T$

• Calorimetry: Measurement of heat flow using a calorimeter.

7.5 Measuring ΔE for Chemical Reactions: Constant-Volume Calorimetry

Constant-volume calorimetry (bomb calorimetry) is used to measure the change in internal energy for reactions at constant volume.

• Bomb Calorimeter: Measures $q_{v}$, the heat at constant volume.

• Application: Used for combustion reactions.

7.6 Enthalpy: The Heat Evolved in a Chemical Reaction at Constant Pressure

Enthalpy (H) is the heat content of a system at constant pressure.

• Enthalpy Change: $\Delta H = \Delta E + P\Delta V$

• Endothermic: Absorbs heat ($\Delta H > 0$); Exothermic: Releases heat ($\Delta H < 0$)

• Application: Used to describe heat flow in chemical reactions at constant pressure.

7.7 Constant-Pressure Calorimetry: Measuring ΔHrxn

Coffee-cup calorimetry measures enthalpy changes for reactions at constant pressure.

• qp = ΔH: At constant pressure, the heat measured equals the enthalpy change.

• Application: Used for reactions in solution.

7.8 Relationships Involving ΔHrxn

Enthalpy changes can be calculated using reaction equations, Hess's Law, and standard enthalpies of formation.

• Hess's Law: The enthalpy change for a reaction is the sum of the enthalpy changes for each step.

• Standard Enthalpy of Formation (ΔHf°): The enthalpy change when one mole of a compound forms from its elements in their standard states.

• Formula: $\Delta H_{rxn}^\circ = \sum n \Delta H_f^\circ (\text{products}) - \sum n \Delta H_f^\circ (\text{reactants})$

Equations and Relationships in Chapter 7

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• First Law: $\Delta E = q + w$

• Work: $w = -P\Delta V$

• Heat: $q = m \times c \times \Delta T$

• Enthalpy: $\Delta H = \Delta E + P\Delta V$

• Hess's Law: $\Delta H_{rxn}^\circ = \sum n \Delta H_f^\circ (\text{products}) - \sum n \Delta H_f^\circ (\text{reactants})$

Chapter 8. The Quantum-Mechanical Model of the Atom

8.2 The Nature of Light

Light exhibits both wave-like and particle-like properties. Understanding light is essential for explaining atomic structure and the periodic table.

• Electromagnetic Radiation: Energy transmitted through space as waves.

• Wavelength (λ), Frequency (ν), Speed (c): $c = \lambda \nu$

• Photoelectric Effect: Demonstrates the particle nature of light; $E = h\nu$

• Electromagnetic Spectrum: Range of all types of electromagnetic radiation.

• Interference and Diffraction: Demonstrate the wave nature of light.

8.3 Atomic Spectroscopy and the Bohr Model

Atomic spectroscopy studies the emission and absorption of light by atoms. The Bohr model explains the line spectrum of hydrogen.

• Bohr Model: Electrons orbit the nucleus in quantized energy levels.

• Energy of Electron in Orbit (Hydrogen): $E_n = -2.18 \times 10^{-18} \left(\frac{1}{n^2}\right)$ J

• Transition Energy: $\Delta E = -2.18 \times 10^{-18} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$ J

8.4 The Wave Nature of Matter: The De Broglie Wavelength, the Uncertainty Principle, and Indeterminacy

Particles such as electrons exhibit wave-like properties, described by the de Broglie wavelength. The uncertainty principle limits the precision of simultaneous measurements of position and momentum.

• De Broglie Wavelength: $\lambda = \frac{h}{mv}$

• Heisenberg Uncertainty Principle: $\Delta x \cdot m \Delta v \geq \frac{h}{4\pi}$

• Application: Explains why electrons do not have definite orbits.

8.5 Quantum Mechanics and the Atom

Quantum mechanics describes the behavior of electrons in atoms using wave functions and quantum numbers.

• Schrödinger Equation: Fundamental equation for wave functions of electrons.

• Quantum Numbers: Principal (n), angular momentum (l), magnetic (ml), and spin (ms).

• Orbitals: Regions of space where electrons are likely to be found.

• Energy Levels: Determined by quantum numbers.

• Atomic Spectroscopy: Energy levels explain the emission and absorption spectra of atoms.

8.6 The Shapes of Atomic Orbitals

Atomic orbitals have characteristic shapes and orientations, described by quantum numbers.

• Probability Density: Probability of finding an electron at a particular location.

• Radial Distribution Function: Probability as a function of distance from the nucleus.

• Nodes: Regions where the probability of finding an electron is zero.

• Shapes: s (spherical), p (dumbbell), d (cloverleaf), f (complex).

• Phase: Sign of the wave function; important in bonding and molecular orbital theory.

Equations and Relationships in Chapter 8

• Speed of Light: $c = \lambda \nu$

• Energy of a Photon: $E = h\nu$

• De Broglie Wavelength: $\lambda = \frac{h}{mv}$

• Heisenberg Uncertainty Principle: $\Delta x \cdot m \Delta v \geq \frac{h}{4\pi}$

• Energy of Electron in Orbit (Hydrogen): $E_n = -2.18 \times 10^{-18} \left(\frac{1}{n^2}\right)$ J

• Transition Energy: $\Delta E = -2.18 \times 10^{-18} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$ J

Equation	Purpose
$PV = nRT$	Relates pressure, volume, temperature, and moles of an ideal gas
$P_{total} = P_1 + P_2 + \ldots$	Dalton's Law of Partial Pressures
$X_i = \frac{n_i}{n_{total}}$	Mole fraction of component i
$KE_{avg} = \frac{3}{2}RT$	Average kinetic energy of a gas particle
$u_{rms} = \sqrt{\frac{3RT}{M}}$	Root mean square velocity of gas molecules
$\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$	Graham's Law of Effusion
$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$	Van der Waals equation for real gases
$\Delta E = q + w$	First Law of Thermodynamics
$q = m \times c \times \Delta T$	Heat transfer calculation
$\Delta H = \Delta E + P\Delta V$	Relationship between enthalpy and internal energy
$\Delta H_{rxn}^\circ = \sum n \Delta H_f^\circ (\text{products}) - \sum n \Delta H_f^\circ (\text{reactants})$	Standard enthalpy of reaction
$c = \lambda \nu$	Relationship between speed, wavelength, and frequency of light
$E = h\nu$	Energy of a photon
$\lambda = \frac{h}{mv}$	De Broglie wavelength
$\Delta x \cdot m \Delta v \geq \frac{h}{4\pi}$	Heisenberg Uncertainty Principle
$E_n = -2.18 \times 10^{-18} \left(\frac{1}{n^2}\right)$ J	Energy of electron in hydrogen atom
$\Delta E = -2.18 \times 10^{-18} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$ J	Energy change for electron transition in hydrogen