User:Prayerfortheworld/formulae

Chapter 3 – Chemical Bonding: The Classical Description
Charge of electron: $$1.602176565 \times 10^{-19} \text{C}$$
 * Constants and Unit conversions

$$1 N = 1 \frac{J}{m}$$

$$1 D = 3.33564 \times 10^{-30} Cm$$

$$\boldsymbol{F} = \frac{q_1q_2}{4\pi\varepsilon_0r^2} = k_e{q_1q_2\over r^2}$$
 * Coulomb's Law

$$(\begin{align} k_e &= \frac{1}{4\pi\varepsilon_0}= 8.988\cdot10^9 \mathrm{N\ m^2\ C}^{-2}) \end{align}$$

$$q_1q_2 = +e \times -e = -e^2$$

$$\boldsymbol{V} = k_e{q_1q_2\over r} = k_e\frac{-e \times +Ze}{r} = -k_e\frac{Ze^2}{r}$$
 * Potential Energy of a pair of charged particles

$$\boldsymbol{F_1} = -\frac{dV}{dr} = -\frac{d}{dr}(-k_e\frac{Ze^2}{r}) = k_e\frac{Ze^2}{r^2}$$
 * Force at point 1 on a potential energy curve

$$\boldsymbol{E_{total}} = {1\over 2} m_ev^2-k_e\frac{Ze^2}{r}$$
 * Total energy of an electron (K+U)

$$V = k_eZe^2(-(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3})+(\frac{1}{r_{12}}+\frac{1}{r_{13}}+\frac{1}{r_{23}}))$$
 * Potential energy for configuration of an element with atomic number $$Z$$ with regards to Coulomb interaction of electrons located at coordinates $$r_1$$, $$r_2$$, $$r_3$$

$$V_{eff}(r) = -k_e\frac{Z_{eff}}{r}$$
 * Effective potential energy function

$$(Z_{eff} = Z - S)$$

$$EN = \frac{1}{2}C(IE_1 + EA)$$
 * Mulliken's scale of electronegativity

Mean of bond dissociation energies: $$\sqrt{\Delta E_{AA}\Delta E_{BB}} $$ Excess bond energy (∆, kJ/mol): $$\Delta = \Delta E_{AB} - \sqrt{\Delta E_{AA}\Delta E_{BB}} $$ Electronegativity difference: $$\chi_A - \chi_B = 0.102\sqrt{\Delta}$$
 * Pauling's scale of electronegativity

$$(\chi_A = EN_A)$$

Given $$\bar{T}$$ is average kinetic energy and $$\bar{V}$$ is average potential energy, and $$\bar{E} = \bar{T} + \bar{V}$$,
 * Virial theorem

$$\bar{T} = -\frac{1}{2}\bar{V}$$

Also, $$\Delta\bar{E} = \Delta\bar{T} + \Delta\bar{V} $$

$$\therefore \Delta\bar{T} = -\frac{1}{2}\Delta\bar{V}$$

$$\And \Delta\bar{E} = \frac{1}{2}\Delta\bar{V}$$

$$\Delta{E_{\infty}} = IE_1(\text{neutral cation}) - EA(\text{neutral anion})$$
 * Coulomb stabilization energy

Energy difference between ionic bond stability and neutral atoms stabilities:

$$\Delta{E_d} \approx -k_e\frac{q_1q_2}{R_e}\frac{N_A}{10^3} - \Delta{E_{\infty}}$$

$$\mu = qR$$
 * Dipole moment

$$\mu = \frac{R}{0.2082 \mbox{Å}} \delta$$

where $$\mu$$, the dipole moment, is in debyes, $$R$$, the bond length, is in angstroms, and $$\delta$$ is the fraction ionic character.

$$\text{Formal charge} = (\text{no. of valence electrons}) - (\text{no. of lone pair electrons}) - \frac{1}{2} (\text{no. of electrons in bonded pairs})$$
 * Formal charge

Chapter 4 – Introduction to Quantum Mechanics
$$c = 2.998 \times 10^{8} m s^{-1}$$
 * Constants

$$h = 6.626 \times 10^{-34} J s$$

$$\text {Bohr radius: } a_0 = \frac{\epsilon_0 h^2}{\pi e^2 m_e} = 5.29 \times 10^{-11} m = 0.529 \text{Å}$$

$$\text{rydberg} = 2.18 \times 10^{-18} J$$

$$\text{1 electronvolt (eV)} = 1.602 \times 10^{-19} J$$

$$\text{mass of electron} = 9.109 \times 10^{-31} kg$$

$$c = \lambda\nu$$
 * Electromagnetic radiation

Amplitude of the electric field:

$$E(x,t) = E_{max} cos(2\pi(\frac{x}{\lambda - \nu t}))$$

Distribution for the intensity of radiation connected with surface oscillators:
 * Blackbody Radiation & Planck's law

$$\rho_{T}(v) =\frac{ 8 \pi h\nu^{3}}{c^3}\frac{1}{ e^{\frac{h\nu}{k_BT}}-1}$$

$$\Delta E = h \nu $$
 * Frequency of light absorbed

$$L = m_evr = n\frac{h}{2\pi}$$ where $$n = 1, 2, 3, ...$$
 * Quantized angular momentum

Allowed values for radius of the orbits:
 * Discrete orbits

$$r_n = \frac{n^2}{Z} a_0$$

Velocity corresponding to the orbit:

$$v_n = \frac{Ze^2}{2\epsilon_0nh}$$

Allowed values for energy:

$$E_n = -(2.18 \times 10^{-18} J) \frac{Z^2}{n^2}$$ where $$n = 1, 2, 3, ...$$

$$E_{max} = \frac{1}{2}mv_e^2 = h\nu - \phi$$
 * Maximum kinetic energy of photoelectrons

$$ n\frac{\lambda}{2} = L $$ where $$n = 1, 2, 3, ...$$
 * Conditions for allowed wavelengths

For standing circular waves:

$$ n\lambda = 2\pi r = L $$ where $$n = 1, 2, 3, ...$$

$$\lambda = \frac{h}{m_ev} = \frac{h}{p}$$
 * Wavelength and linear momentum

$$ (\Delta A)(\Delta B) \geq \frac{h}{4\pi} \longrightarrow (\Delta x)(\Delta p) \geq \frac{h}{4\pi} $$
 * Heisenberg indeterminacy principle


 * Schrödinger's equation

$$E \Psi(\mathbf{r}) = \frac{-h^2}{8\pi^2m}\frac{\partial^2}{\partial x^2} \Psi(\mathbf{r}) + V(\mathbf{r}) \Psi(\mathbf{r})$$

$$E\Psi = \hat H \Psi$$


 * Interpretation of Schrödinger's equation


 * 1) Normalized probability density $$\int_{-\infty}^{+\infty} P(x) dx = \int_{-\infty}^{+\infty} [\psi(x)]^2 dx = 1$$
 * 2) $$P(x)$$ continuous at each point $$x$$ (boundary condition)
 * 3) $$\psi(x)$$ bounded at large values of $$x$$ (boundary condition) $$\psi \rightarrow 0 \text{ as } x \rightarrow \pm \infty$$

Normalized wave function for one-dimensional particle in a box:
 * Particle in a box

$$\psi_n(x) = \sqrt{\frac{2}{L}} sin (\frac{n \pi x}{L})n $$ where $$n = 1, 2, 3, ...$$

Energy for particle described by $$\psi_n(x)$$:

$$E_n = \frac{n^2h^2}{8mL^2}$$

Particle in a box for three-dimensional box:

$$E_{n_xn_yn_z} = \frac{h^2}{8mL^2} [n_x^2 + n_y^2 + n_z^2] $$ where $$n_x, n_y, n_z = 1, 2, 3, ...$$

Chapter 5: Quantum Mechanics and Atomic Structure

 * Quantum numbers

Chapter 6: Quantum Mechanics and Molecular Structure
Electrons approach each other, approximate wave function:
 * VB theory

$$\psi^{el}(r_{1A}, r_{2B}; R_{AB}) = c_1(R_{AB})\phi^A(r_{1A})\phi^B(r_{2B})$$

Atoms interact, wave function:

$$\psi^{el}(r_{1A}, r_{2B}; R_{AB}) = c_1(R_{AB})\phi^A(r_{1A})\phi^B(r_{2B}) + c_2(R_{AB})\phi^A(r_{2A})\phi^B(r_{1B})$$

$$\psi^{el}_g = c_1(R_{AB})\phi^A(r_{1A})\phi^B(r_{2B}) + c_2(R_{AB})\phi^A(r_{2A})\phi^B(r_{1B})$$

$$\psi^{el}_u = c_1(R_{AB})\phi^A(r_{1A})\phi^B(r_{2B}) - c_2(R_{AB})\phi^A(r_{2A})\phi^B(r_{1B})$$

Wave function for electron pair bond:

$$\psi^{el}_g = C_1[1s^A(1)1s^B(2) + 1s^A(2)1s^B(1)]$$

Chapter 3 – Chemical Bonding: The Classical Description
Repulsive: $$\vec{F}_{12} = k_e{|q_1q_2|\over r^2}\vec{r}_{12}$$
 * Force on $$q_2$$ due to $$q_1$$

Attractive: $$\vec{F}_{12} = -\vec{F}_{21}$$

Charge 1 creates electric field that exerts force on charge 2
 * Interpretation of $$\vec{F}_{12}$$

$$\vec{E} = \lim_{q_0 \to 0}\frac{\vec{F}}{q_0} = k\frac{q}{r^2}\vec{r}$$

$$\therefore \vec{F} = q\vec{E}$$

Charge $$q_0$$ acts as probe of electric ﬁeld (and force) due to $$q$$

$$\vec{F} = q_0\vec{E}(x) = k_e\frac{q_0q}{x^2}\vec{x}$$
 * Force on probe $$q_0$$

Potential energy is equivalent to work:

$$U \equiv W = \int_{-\infty}^{x_0}{q_0E(x') dx'}$$

After integration,

$$U = |q_0|k_e\frac{q}{x} = |q_0|V(x)$$

Where $$V(x)$$ is the Coulomb potential at x due to charge $$q$$

$$\vec{E} = -\frac{d}{dx}V(x)\vec(x)$$

Chapter 4 – Introduction to Quantum Mechanics
If energy is quantized as $$E = h f$$, where $$h$$ is Planck's constant and $$f$$ ($$\nu_e$$ in the textbook) is the frequency of the incident photon.
 * Maximum kinetic energy of an ejected electron

$$K_{max} = h f - \phi$$

$$\phi$$ is the work function $$W$$, and $$f_0$$ is the threshold frequency.

$$\phi = W = h f_0$$

$$\therefore K_{max} = h f - h f_0 = h (f - f_0)$$

Photoelectric effect will occur if $$f > f_0$$ (kinetic energy is positive).

$$K_{max} = h f - \phi$$ can be used to determine graphical representation of kinetic energy; $$h$$ is the slope, $$-\phi$$ is the y-intercept, and $$f_0$$ is the x-intercept.