Gas Laws: Boyle's, Charles's, and Avogadro's

Gas Properties

• Gases are highly compressible
• The volume of a gas varies with container size
• Gases expand to fill their container
• Gases form homogeneous mixtures

Pressure

• Pressure is defined as force per unit area: $P = \frac{F}{A}$
• The SI unit for pressure is the Pascal (Pa): $1 Pa = 1 N/m^2$
• Other units for pressure include mmHg, torr, atm, and kPa
• Conversions: $1 atm = 760 mmHg = 760 torr = 1.01325 \times 10^5 Pa = 101.325 kPa$

Boyle's Law

• States that volume is inversely proportional to pressure at constant temperature and number of moles: $V \propto \frac{1}{P}$
• Formula: $P_1V_1 = P_2V_2$

Charles's Law

• States that volume is directly proportional to temperature at constant pressure and number of moles: $V \propto T$
• Formula: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Avogadro's Law

• States that volume is directly proportional to the number of moles at constant temperature and pressure: $V \propto n$
• Formula: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

Ideal-Gas Equation

• Relates pressure, volume, number of moles, and temperature: $PV = nRT$
• $R$ is the ideal gas constant, with values of $0.08206 \frac{L \cdot atm}{mol \cdot K}$ or $8.314 \frac{J}{mol \cdot K}$

Standard Temperature and Pressure (STP)

• Defined as $0^\circ C$ (273.15 K) and 1 atm
• The molar volume of an ideal gas at STP is 22.41 L

Gas Density

• Can be calculated using the formula: $d = \frac{m}{V} = \frac{MP}{RT}$, where $M$ is molar mass

Molar Mass

• Can be calculated using the formula: $M = \frac{dRT}{P}$

Dalton's Law of Partial Pressures

• States that the total pressure of a gas mixture is the sum of the partial pressures of each component: $P_t = P_1 + P_2 + P_3 +...$

Partial Pressure

• Defined as the mole fraction of gas $i$ times the total pressure: $P_i = X_i P_t$
• Mole fraction is defined as: $X_i = \frac{n_i}{n_t}$

Gas Stoichiometry

• The ideal-gas equation can be used to relate volume, pressure, temperature, and moles in chemical reactions.

Kinetic-Molecular Theory of Gases : Postulates

• Gases consist of particles in continuous, random motion.
• The volume of gas particles is negligible compared to the total volume.
• Attractive/repulsive forces between particles are negligible.
• Average kinetic energy is proportional to absolute temperature.
• Collisions are elastic (energy can be transferred, but total energy remains constant).

Average Kinetic Energy

• The formula for calculating average kinetic energy: $\bar{KE} = \frac{3}{2}RT$

Root-Mean-Square (rms) Speed

• The formula for calculating rms speed: $u = \sqrt{\frac{3RT}{M}}$

Effusion

• The escape of gas molecules through a tiny hole.

Graham's Law

• States that the rate of effusion or diffusion is inversely proportional to the square root of the molar mass: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

Diffusion

• The spread of one substance through another.

Real Gases

• Deviate from ideal behavior at high pressure and low temperature

van der Waals Equation

• Corrects ideal gas behavior for real gases: $(P + \frac{an^2}{V^2})(V - nb) = nRT$
• $a$ corrects for intermolecular attractions
• $b$ corrects for the volume of gas molecules

Radiative Heat Transfer

• Heat transfer occurs by conduction, convection, and radiation.
• Conduction is energy transfer through particle interaction within a substance.
• Convection is energy transfer between a solid surface and a moving fluid.
• Radiation is energy emitted as electromagnetic waves due to changes in electronic configurations.
• Conduction and convection require a medium.

Radiation does not

• Radiation doesn't need a medium
• Radiation travels at the speed of light in a vacuum.
• All bodies at a temperature above absolute zero emit thermal radiation.

Blackbody Radiation

• Blackbody is a perfect emitter and absorber of radiation.
• Blackbody emits maximum energy at a given temperature and wavelength.
• Blackbody absorbs all incident radiation.

Blackbody Emissive Power

• Blackbody emissive power: $E_b(T) = \sigma T^4$
• $\sigma = 5.67 \times 10^{-8} W/m^2 \cdot K^4$: Stefan-Boltzmann constant $T$: absolute temperature in Kelvin (K)

Emissivity

• Emissivity represents how close a surface is to a blackbody. Denoted by $\epsilon$.
• Emissivity ranges between between 0 and 1: ($0 \le \epsilon \le 1$)
• Blackbodies have an emissivity of 1.

Molecular Orbitals

LCAO

• Molecular orbitals are approximated by linear combinations of atomic orbitals
• $$\phi = \sum_{i}^{n}c_i\chi_i$$
• $\phi$: molecular orbital
• $\chi_i$: atomic orbitals
• $c_i$: coefficients
• $n$: number of atomic orbitals

$\mathrm{H}_2^+$: Schrödinger equation

• The schrödinger equation $\hat{H}\psi = E\psi$
• Can also be written as: $\left(-\frac{\hbar^2}{2m}\nabla^2 + V \right)\psi = E\psi$

$\mathrm{H}_2^+$

• The schrödinger equation: $$\left(-\frac{\hbar^2}{2m}\nabla^2 - \frac{e^2}{4\pi\epsilon_0r_A} - \frac{e^2}{4\pi\epsilon_0r_B} + \frac{e^2}{4\pi\epsilon_0R} \right)\psi = E\psi$$

Born-Oppenheimer approximation

• Nuclear kinetic energy is negligible compared to that of the electron

Trial wave function

• $$\psi = c_A\chi_A + c_B\chi_B$$
• $\chi_A$: 1s orbital on atom A
• $\chi_B$: 1s orbital on atom B

Secular equation

$$\begin{vmatrix} H_{AA} - ES_{AA} & H_{AB} - ES_{AB} \ H_{BA} - ES_{BA} & H_{BB} - ES_{BB} \end{vmatrix} = 0$$

Secular determinant

$$H_{ij} = \int \chi_i^* \hat{H} \chi_j d\tau$$

$$S_{ij} = \int \chi_i^* \chi_j d\tau$$

Energy levels

$$E_{\pm} = \frac{H_{AA} \pm H_{AB}}{1 \pm S_{AB}}$$

• $H_{AA} = H_{BB} = \int \chi_A^* \hat{H} \chi_A d\tau$ : Coulomb integral
• $H_{AB} = H_{BA} = \int \chi_A^* \hat{H} \chi_B d\tau$ : Resonance integral
• $S_{AB} = \int \chi_A^* \chi_B d\tau$ : Overlap integral

Wave functions

$$\psi_+ = \frac{1}{\sqrt{2 + 2S_{AB}}}(\chi_A + \chi_B)$$

$$\psi_- = \frac{1}{\sqrt{2 - 2S_{AB}}}(\chi_A - \chi_B)$$

Bonding and Antibonding Orbitals

• $\psi_+$: bonding orbital
• $\psi_-$: antibonding orbital

1s$\sigma_g$ and 1s$\sigma_u^*$

• $\sigma$: $\lambda = 0$ cylindrically symmetrical around the internuclear axis
• g: gerade, symmetric with respect to inversion through the center of the molecule
• u: ungerade, antisymmetric with respect to inversion through the center of the molecule
• * : antibonding