Equations and Constants in Physics and Chemistry

Key Equations and Constants

Pressure relationships:
$PH = P0 e^{- rac{MgH}{RT}}$

• Units: Pressure (P) in Pascals (Pa), Height (H) in meters (m), Mass (M) in kilograms (kg), Gravity (g) in m/s², and Temperature (T) in Kelvin (K).

• Usage: Used in calculating pressure at different altitudes.

$PV = nRT$

• Units: Pressure (P) in atm or Pa, Volume (V) in liters (L), Number of moles (n), Ideal gas constant (R) in specific units, Temperature (T) in Kelvin (K).

• Usage: This equation is used in gas law calculations involving ideal gases.

Constants:
Avogadro’s constant: $6.02 \times 10^{23} \text{ mol}^{-1}$

• Usage: Used to relate the number of particles to moles.

Ideal gas constant:
$R = 8.314 \text{ kg m}^2 \text{s}^{-2} \text{K}^{-1} \text{mol}^{-1}$
$R = 8.206 \times 10^{-2} \text{ L atm K}^{-1} \text{mol}^{-1}$

• Usage: Used in the ideal gas law and related calculations.

Temperature Conversion:
$0 \, ^\circ C = 273.15 \, K$

• Usage: Used to convert Celsius to Kelvin for thermodynamic equations.

Properties of air:
Average molar mass: $28.97 \, \text{g mol}^{-1}$

• Usage: Helps in calculations involving air mixtures and gas laws.

Density at 25 °C/1 atm: $1.2 \times 10^{3} \text{ g m}^{-3}$

• Usage: Used for determining the mass of air in a given volume under specified conditions.

Viscosity at 25 °C/1 atm: $1.9 \times 10^{-2} \text{ g m}^{-1} \text{s}^{-1}$

• Usage: Used for fluid dynamics calculations involving air.

Acceleration due to gravity:
$g = 9.81 \, \text{m s}^{-2}$

• Usage: Used in calculating forces due to gravity.

Unit conversions:
$1 \, \text{km}^2 = 10^6 \, \text{m}^2$
$1 \, \text{Pa} = 1 \, \text{N m}^{-2}$
$1 \, \text{N} = 1 \, \text{kg m s}^{-2}$

• Usage: Helpful in converting between different units in equations.

Force and Pressure equations:
Force formula:
$F = m \times a$

• Units: Force (F) in Newtons (N), Mass (m) in kg, Acceleration (a) in m/s².

• Usage: Used to calculate the force acting on an object.

Pressure formula:
$P = \frac{\text{Force}}{\text{Area}}$

• Units: Pressure (P) in Pascals (Pa), Force in Newtons (N), Area in square meters (m²).

• Usage: Used to calculate pressure applied over a surface.

Pressure Units:
$1 \, \text{atm} = 101,325 \, \text{Pa}$

• Usage: Conversion between common pressure units.

Radioactivity:
$1 \, \text{pCi} = 0.037 \, \text{Bq}$

• Usage: Used to convert between different measurements of radioactivity.

Number of moles:
$Nt = N0 e^{-\lambda t}$

• Units: N (Number of particles/nuclei), t in time (s).

• Usage: Used in decay calculations to find the remaining quantity of a radioactive substance.

Half-life equation:
$t_{1/2} = \frac{0.693}{\lambda}$

• Units: Time (t) in seconds.

• Usage: Useful in calculating the half-life of substances.

Chemical Equilibrium:
$K<em>{a1} = \frac{[H^{+}][HCO3^{-}]}{[H2CO3]} = 4.45 \times 10^{-7}$ $K</em>{a2} = \frac{[H^{+}][CO3^{2-}]}{[HCO3^{-}]} = 4.69 \times 10^{-11}$

• Units: Equilibrium constant (K) is dimensionless.

• Usage: Used in equilibrium calculations to determine concentrations at equilibrium.

pH and pOH Calculations:
$pOH = -\log [OH^{-}]$
$pH = -\log [H^{+}]$

• Units: pH and pOH are unitless.

• Usage: Used in acid-base chemistry to measure the concentration of hydrogen ions.

Ion product constant:
$K_w = 1.0 \times 10^{-14} \, \text{at} \, 25 \, ^\circ C$

• Usage: Used in calculations related to water ionization and acidity.

Electrochemistry:
Standard reduction potential:
$E_{red}^{\circ} = 0.0591 \, V$

• Units: Voltage (V).

• Usage: Used in determining the feasibility of redox reactions.

at 25 °C:
$pE = pE^{\circ} - \frac{1}{n} \log Q$
where $Q$ is the reaction quotient.

• Usage: Used in calculating electrochemical cell potentials.

Gaseous solubility:
$S<em>{gas} = K</em>H P_{gas}$

• Units: Solubility ($S<em>{gas}$) often in mol/L, and pressure ($P</em>{gas}$) in atm or Pa.

• Usage: Used in determining how much gas can dissolve into a liquid under given pressure conditions.

$K<em>{ow} = \frac{[S]</em>{o}}{[S]_{w}}$

• Usage: Used in environmental chemistry to understand the partitioning of chemicals between water and organic phases.

Wave Properties:
Energy of photons:
$E = h v$

• Units: Energy (E) in joules (J), frequency (v) in Hz.

• Usage: Used to relate the energy of photons to their frequency.

Speed of light:
$c = \lambda \times v$
$c = 2.998 \times 10^8 \, \text{m s}^{-1}$

• Units: Speed (c) in m/s, wavelength ($\lambda$) in meters, frequency (v) in Hz.

• Usage: Used in optical physics to calculate the relationship between light's speed, wavelength, and frequency.

Planck's Constant:
$h = 6.626 \times 10^{-34} \, J.s$

• Usage: Fundamental constant for quantum mechanics and photon energy calculations.

Area Formulas:
Circle: $A = \pi r^2$

• Units: Area (A) in m², radius (r) in m.

• Usage: Used to calculate the area of circles.

Sphere: $A = 4\pi r^2$

• Usage: Used to calculate the surface area of spheres.

Thermodynamics:
Stefan-Boltzmann law:
$E_{total} = \sigma T^4$
where $\sigma = 5.67 \times 10^{-8} \, \text{W m}^{-2} K^{-4}$

• Units: Total energy (E) in W/m², temperature (T) in K.

• Usage: Used in climate science to calculate the total energy radiated by a black body.

Logarithmic Properties:
$\log \frac{x}{y} = \log x - \log y$
$\log xy = y \log x$
$\log x^y = y \log x$

• Usage: Used in various mathematical calculations involving logarithms.

Changing Conditions:
$\ln \left( \frac{c2}{c1} \right) = \frac{\Delta H}{R} \left( \frac{1}{T1} - \frac{1}{T2} \right)$

• Units: Concentration (c) units vary, enthalpy change (ΔH) in J/mol, temperature (T) in K.

• Usage: Used to find out how concentration changes with temperature when heat is added or removed.

Fluid Dynamics Equation:
$vt = \frac{(\rho<em>p - \rho</em>a) C g d}{2 \eta}$
where:
$\rho<em>p$: density of solid $\rho</em>a$: density of air
$C$: constant related to fluid viscosity
$g$: acceleration due to gravity
$d$: diameter of the particle
$\eta$: viscosity

• Units: Velocity (v) in m/s, density (ρ) in kg/m³, diameter (d) in m, viscosity (η) in Pa·s.

• Usage: Used in fluid dynamics to find the terminal velocity of particles in a fluid.