Exhaustive University Physics and Chemistry Formulae Chemistry Study Formulas and Chemistry Study Notes

Principles of Light and Optics

Geometrical optics and wave behavior are foundational to understanding how light interacts with different media. Snell's Law describes the path of refracted light as it passes between two substances with different indices of refraction. It is mathematically expressed as:

$n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

In this equation, $n_1$ and $n_2$ represent the indices of refraction for the first and second media, respectively, while $\theta_1$ is the incident angle and $\theta_2$ is the refracted angle relative to the normal. The index of refraction ($n$) for any given medium is defined by the ratio of the speed of light in a vacuum ($c$) to the speed of light in that medium ($v$):

$n = \frac{c}{v}$

Light also exhibits particle-like behavior through photons. The energy of a photon is quantified using the Planck Relation, which relates energy ($E$) to frequency ($f$) and wavelength ($\lambda$). This is governed by Planck's constant ($h = 6.626 \times 10^{-34}\,\text{J}\cdot\text{s}$) and the speed of light ($c$):

$E = hf = \frac{hc}{\lambda}$

For mirrors and lenses, the Lensmaker's Equation and magnification formulas are essential. The relationship between the focal length ($f$), the distance between the object and the lens/mirror ($d_o$), and the distance between the image and the lens/mirror ($d_i$) is given by:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

The magnification ($m$) describes the ratio of the image distance to the object distance. A magnification of magnitude 1 indicates the image is the same size as the object, while $m > 1$ denotes an enlarged image and $m < 1$ indicates a reduced image:

$m = -\frac{d_i}{d_o}$

Optical power is measured in diopters ($D$), which is the reciprocal of the focal length in meters:

$P = \frac{1}{f}$

Wave interference and diffraction are also critical. In single-slit diffraction, the positions of dark fringes (minima) are determined by the width of the slit ($a$), the wavelength of the incident wave ($\lambda$), and the angle ($\theta$):

$a \sin(\theta) = n\lambda$

Electricity and Magnetism

Electrostatics starts with Coulomb's Law, which determines the magnitude of the electrostatic force ($F_e$) between two point charges ($q_1$ and $q_2$) separated by a distance ($r$):

$F_e = \frac{k |q_1| |q_2|}{r^2}$

Coulomb's constant ($k$) is approximately $8.99 \times 10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$. The electric field ($E$) generated by a source charge ($Q$) acting on a test charge ($q$) is given by:

$E = \frac{F_e}{q} = \frac{kQ}{r^2}$

Electric potential energy ($U$) and electric potential ($V$) are related to the work required to move a charge within an electric field. Their formulas are as follows:

$U = \frac{kQq}{r} = q\Delta V = qEd$

$V = \frac{kq}{r}$

The potential difference, or voltage ($\Delta V$), is the change in potential between two points, expressed in Volts ($V$), which are equivalent to Joules per Coulomb ($\text{J/C}$). Current ($I$) is defined as the flow of charge ($Q$) over a specific interval of time ($t$):

$I = \frac{Q}{t}$

Ohm's Law provides the fundamental relationship between voltage ($V$), current ($I$), and resistance ($R$):

$V = IR$

Electric circuits can be arranged in series or parallel. In series circuits, the equivalent resistance ($R_{eq}$) is the sum of individual resistances, while the current remains constant across all components, and voltage is additive:

$R_{eq} = R_1 + R_2 + \dots$

$I_{total} = I_1 = I_2 = \dots$

$V_{total} = V_1 + V_2 + \dots$

In parallel circuits, the voltage drop is equal across all branches, while total current is the sum of branch currents, and the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances:

$V_{total} = V_1 = V_2 = \dots$

$I_{total} = I_1 + I_2 + \dots$

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$

Capacitance ($C$) measures the ability of a component to store charge per unit voltage. The energy stored by a capacitor ($U$) can be calculated in multiple ways:

$C = \frac{Q}{V}$

$U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C}$

Capacitors in parallel add directly ($C_{eq} = C_1 + C_2 + \dots$), whereas capacitors in series follow the reciprocal rule ($\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$). Electric power ($P$) dissipated by a resistor is calculated through:

$P = IV = \frac{V^2}{R} = I^2R$

Solutions and Equilibrium

Chemical equilibrium for a reaction is governed by the Law of Mass Action, which relates the concentrations of products and reactants to an equilibrium constant ($K_{eq}$):

$K_{eq} = \frac{[C]^c [D]^d}{[A]^a [B]^b}$

If $K_{eq} > 1$, the reaction favors products; if $K_{eq} < 1$, it favors reactants; and if $K_{eq} \approx 1$, products and reactants are roughly equal. Acids and bases are handled using the Henderson-Hasselbalch equation to find pH or pOH:

$pH = pK_a + \log\left(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\right)$

$pOH = pK_b + \log\left(\frac{[\text{conjugate acid}]}{[\text{weak base}]}\right)$

Additional pH/pOH relationships include:

$pH = -\log([H^+])$

$pOH = -\log([OH^-])$

$pH + pOH = 14$

Dissociation constants for acids and bases are defined as:

$K_a = \frac{[H_3O^+][A^-]}{[HA]}$

$K_b = \frac{[B^+][OH^-]}{[BOH]}$

Osmotic pressure ($\Pi$) in solutions identifies the pressure needed to prevent osmosis and depends on the Van't Hoff factor ($i$), molar concentration ($M$), the ideal gas constant ($R$), and temperature ($T$):

$\Pi = iMRT$

Thermodynamics and Thermochemistry

Thermal expansion occurs when matter changes its dimensions due to temperature flux. For solids (linear expansion) and liquids/solids (volumetric expansion), the changes are determined by:

$\Delta L = \alpha L\Delta T$

$\Delta V = \beta V\Delta T$

The transfer of heat ($Q$) without a phase change is given by $Q = mc\Delta T$, where $c$ is specific heat (for water, $c = 4.184\,\text{J/g}\cdot\text{K}$). During a phase change, heat is calculated using latent heat ($L$):

$Q = mL$

The first law of thermodynamics states that the change in internal energy ($\Delta U$) is the heat added to the system ($Q$) minus the work done by the system ($W$):

$\Delta U = Q - W$

For an ideal gas, internal energy is function of moles ($n$) and temperature ($T$):

$U = \frac{3}{2}nRT$

Reaction energetics are defined by Gibbs Free Energy ($\Delta G$) and Enthalpy ($\Delta H$). The relationship between free energy, enthalpy, and entropy ($\Delta S$) is:

$\Delta G = \Delta H - T\Delta S$

The standard heat of a reaction ($\Delta H_{rxn}$) can be calculated from formation or bond enthalpies:

$\Delta H_{rxn}^0 = \sum \Delta H_{products}^0 - \sum \Delta H_{reactants}^0$

$\Delta H_{rxn} = \sum \Delta H_{\text{bonds broken}} - \sum \Delta H_{\text{bonds formed}}$

In electrochemistry, the electromotive force ($emf$) and Gibbs Free Energy are linked by Faraday's constant ($F = 96,485\,\text{C/mol } e^-$):

$emf = E_{\text{cathode}}^0 - E_{\text{anode}}^0$

$\Delta G = -nFE_{\text{cell}}^0$

Gasses and Fluids

The Ideal Gas Law combines several variables: pressure ($P$), volume ($V$), moles ($n$), and temperature ($T$):

$PV = nRT$

Alternative gas laws include:

• Boyle's Law: $PV = k$
• Charles's Law: $\frac{V}{T} = k$
• Gay-Lussac's Law: $\frac{P}{T} = k$
• Avogadro's Principle: $\frac{n}{V} = k$
• Combined Gas Law: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

Fluid dynamics explores density ($\rho = m/V$) and pressure ($P = F/A$). Absolute pressure in a fluid depth ($h$) is:

$P = P_0 + \rho gh$

Archimedes' Principle states the buoyant force ($F_{buoy}$) equals the weight of the fluid displaced:

$F_{buoy} = \rho_{\text{fluid}} V_{\text{fluid displaced}} g$

Pascal's Principle states pressure applied to an enclosed fluid is transmitted undiminished:

$P = \frac{F_1}{A_1} = \frac{F_2}{A_2}$

Poiseuille's Law describes the volume flow rate ($Q$) of a viscous fluid through a pipe:

$Q = \frac{\pi r^4 \Delta P}{8\eta L}$

Bernoulli's Equation expresses the conservation of energy in flowing fluids:

$P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$

Mechanics and Kinematics

Linear motion is described by kinematics equations involving displacement ($d$), velocity ($v$), acceleration ($a$), and time ($t$):

$v = v_0 + at$

$d = v_0 t + \frac{1}{2}at^2$

$v^2 = v_0^2 + 2ad$

$d = \frac{1}{2}(v_0 + v)t$

Projectile motion components are derived via trigonometric functions:

$v_{0,x} = v_0 \cos(\theta)$

$v_{0,y} = v_0 \sin(\theta)$

Dynamics focuses on Force ($F = ma$) and Newton's Universal Law of Gravitation:

$F_{grav} = \frac{GMm}{r^2}$

where $G = 6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$. Static and kinetic friction are given by:

$F_{s,max} = \mu_s F_N$

$F_k = \mu_k F_N$

On an inclined plane, the gravitational force components are:

• Parallel: $F = mg \sin(\theta)$
• Perpendicular: $F = mg \cos(\theta)$

Torque ($\tau$) and Centripetal Force ($F_c$) define rotational and circular motions:

$\tau = rF\sin(\theta) = rF$

$F_c = \frac{mv^2}{r}$

$a_c = \frac{v^2}{r}$

Energy forms include Kinetic Energy ($KE = \frac{1}{2}mv^2$), Gravitational Potential Energy ($U = mgh$), and Elastic Potential Energy ($U = \frac{1}{2}kx^2$). The Work-Energy Theorem states that Net Work ($W_{net}$) equals the change in kinetic energy:

$W_{net} = \Delta KE = KE_f - KE_i$

Work itself is the product of force and displacement:

$W = Fd\cos(\theta)$

Power ($P$) is the rate of energy transfer:

$P = \frac{W}{t} = \frac{\Delta E}{t}$

Mechanical Advantage ($MA$) and Efficiency compare output forces and work to inputs:

$MA = \frac{F_{resistance}}{F_{effort}}$

$\text{Efficiency} = \frac{W_{out}}{W_{in}} = \frac{(\text{load})(\text{load distance})}{(\text{effort})(\text{effort distance})} \times 100\%$

Waves and Sound

Wave propagation speed ($v$) is the product of wavelength and frequency:

$v = \lambda f$

Period ($T$) is the reciprocal of frequency ($T = 1/f$). The Doppler Effect describes the change in perceived frequency due to relative motion between detector and source:

$f_{perceived} = f_{actual} \left(\frac{v \pm v_d}{v \mp v_s}\right)$

Sound intensity level ($\beta$) is measured in decibels ($dB$) relative to the threshold of hearing ($I_0 = 1 \times 10^{-12}\,\text{W/m}^2$):

$\beta = 10 \log\left(\frac{I}{I_0}\right)$

Standing waves in pipes are determined by the harmonic ($n$) and the length of the pipe ($L$):

• Open pipes: $\lambda = \frac{2L}{n}$, $f = \frac{nv}{2L}$
• Closed pipes (one end): $\lambda = \frac{4L}{n}$, $f = \frac{nv}{4L}$ (where $n$ is an odd integer)

Beat frequency when two sounds are close in frequency is given by:

$f_{beat} = |f_1 - f_2|$