Equations
Temperature & Unit Conversions
Converts temperature from Celsius to Kelvin.
\[ T/\text{K} = t/^\circ\text{C} + 273.15 \]
Converts temperature from Fahrenheit to Celsius.
\[ t/^\circ\text{C} = (t/^\circ\text{F} - 32)/1.8 \]
Converts temperature from Celsius to Fahrenheit.
\[ t/^\circ\mathrm{F} = (1.8 \times t/^\circ\mathrm{C}) + 32 \]
Fundamental Quantitative Concepts
Relates the mass of a substance to the volume it occupies.
\[ \rho = \frac{m}{V} \]
Calculated from the fractional abundances and masses of an element’s isotopes (AE).
\[ A_{\mathrm{r}}(\text{E}) = \sum_i ~ \left [x(^{A}\text{E}) ~ A_{\mathrm{r}}(^{A}\text{E}) \right ]_i \]
Calculated from the standard atomic weights of the atoms making up the entity (X)
\[ M_{\mathrm{r}}(\mathrm{X}) = \sum_i ~ \left [ N \times A_{\mathrm{r}}{^{\circ}}(\text{E}) \right ]_i \]
Calculated from the mass of entity divided by the number of moles of X.
or from the relative molecular mass multiplied by the molar mass constant.
\[ M(\mathrm{X}) = M_{\mathrm{r}}(\mathrm{X}) ~ M_{\mathrm{u}} \]
The moles of a substance is equal to its mass divided by its molar mass.
\[ n = \frac{m}{M} \]
The contribution of one element to the total mass of a compound.
where mass fraction, w(E), can be
\[ \begin{align*} w(\mathrm{E}) &= \dfrac{m(\mathrm{E})}{m(\mathrm{X})} \\[1.5ex] &= \dfrac{N \times A_{\mathrm{r}}{^{\circ}}}{M_{\mathrm{r}}(\mathrm{X})} \end{align*} \]
Compares the actual experimental yield to the maximum theoretical yield.
\[ \mathrm{\%~yield} = \frac{\text{actual yield}}{\text{theoretical yield}} \times 100\% \]
Quantum & Atomic Structure
Relates the energy of a single photon to its frequency.
\[ E = h\nu \]
Relates the speed of light to its wavelength and frequency.
\[ c = \lambda\nu \]
Relates photon energy directly to its wavelength.
\[ E = \frac{hc}{\lambda} \]
Calculates the wavelength of a particle based on its momentum.
\[ \lambda = \frac{h}{mv} \]
Calculates the energy change of an electron transition in a hydrogen atom.
\[ \Delta E = -2.18 \times 10^{-18} \text{ J}~\left(\frac{1}{n_f{^{2}}} - \frac{1}{n_i{^{2}}}\right) \]
Describes the kinetic energy of an electron ejected by a photon.
\[ \text{KE} = h\nu - \Phi \]
States the fundamental limit to the precision of measuring position and momentum.
\[ \Delta x \cdot m\Delta v \ge \frac{h}{4\pi} \]
Gas Laws & Behavior
Relates pressure and volume at constant temperature and moles.
\[ P_1V_1 = P_2V_2 \]
Relates volume and absolute temperature at constant pressure and moles.
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Relates pressure and absolute temperature at constant volume and moles.
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
Relates volume and moles at constant pressure and temperature.
\[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \]
Combines Boyle’s, Charles’s, and Gay-Lussac’s laws for a fixed amount of gas.
\[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]
Relates pressure, volume, moles, and temperature for an ideal gas.
\[ PV = nRT \]
The total pressure of a gas mixture is the sum of the partial pressures of its components.
\[ P_\text{total} = P_1 + P_2 + P_3 + \dots \]
Calculates molar mass from the properties of a gas.
\[ M = \frac{mRT}{PV} \]
Calculates the density of a gas from its properties.
\[ d = \frac{MP}{RT} \]
Calculates the average speed of gas particles based on temperature and molar mass.
\[ \mu_\text{rms} = \sqrt{\frac{3RT}{M}} \]
Compares the rates of effusion of two gases based on their molar masses.
\[ \frac{\text{rate of effusion A}}{\text{rate of effusion B}} = \sqrt{\frac{\text{MW}_B}{\text{MW}_A}} \]
Adjusts the ideal gas law to account for real gas behavior (particle volume and intermolecular attractions).
\[ \left(P + a~\frac{n^2}{V^2}\right)(V-nb) = nRT \]
Compositions of Mixtures
Fractions
The ratio of the mass of a specific component (i) to the total mass of the entire mixture. This value is dimensionless and is often expressed as a percentage.
\[ w_i = \dfrac{m_i}{m_{\mathrm{total}}} \]
A mass fraction (w) expressed as a percentage
\[ w_i~\% = w_i \times 100\% \]
The ratio of the volume of a specific component (i) before mixing to the total volume of the entire mixture. This is a common unit for liquid-liquid solutions.
\[ \phi_i = \dfrac{V_i}{V_{\mathrm{total}}} \]
Volume fraction (φ) expressed as a percentage
\[ \phi_i~\% =\phi_i \times 100~\% \]
The ratio of the number of moles of a specific component (i) to the total moles of the entire mixture. This value is dimensionless and is often expressed as a percentage.
\[ x_i = \dfrac{n_i}{n_{\mathrm{total}}} \]
Mole fraction (xi) expressed as a percentage
\[ x_i~\% = x_i \times 100~\% \]
Ratios
A ratio of the number of entities (atoms, molecules, ions, etc.) of one substance (B) to the number of entities of another substance
\[ r(\mathrm{B,A}) = \dfrac{N_{\mathrm{B}}}{N_{\mathrm{A}}} \]
A ratio of the amount of substance (in moles) of one substance (B) to the amount of another substance (A).
\[ r(\mathrm{B,A}) = \dfrac{n_{\mathrm{B}}}{n_{\mathrm{A}}} \]
Concentrations
Relates the mass of a solute to the total volume of the solution.
\[ \rho_i = \dfrac{m_i}{V} \]
Relates the volume of a constituent to the total volume of the mixture.
\[ \sigma_i = \dfrac{V_i}{V} \]
Relates the number of entities of a constituent to the total volume of the mixture.
\[ C = \dfrac{N_{\mathrm{B}}}{V} \]
The formal definition of concentration, relating the amount of solute to the total volume of the solution (typically mol L−1).
\[ c_{\mathrm{B}} = \dfrac{n_{\mathrm{B}}}{V} \]
The common practical unit for amount concentration, defined as moles of solute per liter of solution.
\[ \class{mjx-molar}{\mathrm{M}} \equiv \dfrac{\mathrm{mol}}{\mathrm{L}} \]
Molality
Defines the amount of solute entities divided by the mass of the solvent (typically mol kg−1) and is temperature independent. Sometimes symbolized with b to avoid confusion with mass (m).
\[ m_{\mathrm{B}} = \dfrac{n_{\mathrm{B}}}{m_{\mathrm{solvent}}} \]
Solubility
A practical way to express the mass of a solute that can dissolve in a specific volume of solvent at a given temperature.
\[ \mathrm{g/100~g} = \left( \dfrac{m_{\mathrm{solute}}}{m_{\mathrm{solvent}}}\right ) \times 100 \]
Expresses solubility as the amount concentration of the solute in a saturated solution at a specific temperature (typically mol L−1).
\[ c_{sat} = \dfrac{n_{\mathrm{solute}}}{V_{\mathrm{solution}}} \]
Solutions & Colligative Properties
Calculates the new molar concentration or volume when a stock solution is diluted.
\[ c_1V_1 = c_2V_2 \]
Relates dissolved gas amount concentration (molarity) to its partial pressure above the solution.
\[ c_g = k_H P_g \]
Calculates vapor pressure of a solution with a non-volatile solute.
\[ P_\text{solution} = x_\text{solvent} P^\circ_\text{solvent} \]
Calculates total pressure of a solution with multiple volatile components.
\[ P_\text{solution} = \sum P_j = \sum x_j P_j{^{\circ}} \]
Relates the vapor pressure of a liquid to its vapor pressure at a specific temperature.
\[ P = A~e^{-\Delta_{\mathrm{vap}} H/RT} \]
Relates the vapor pressure of a liquid to its vapor pressure at a specific temperature.
\[ \ln P = -\dfrac{\Delta_{\mathrm{vap}}H}{R}\left ( \dfrac{1}{T} \right ) + \ln A \]
A rearrangement of the equation for a line. Determines the enthalpy of vaporization of a liquid via two data points of vapor pressure and temperature.
\[ \ln\left(\frac{P_2}{P_1}\right) = -\frac{\Delta_{\mathrm{vap}} H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]
The decrease in a solvent’s freezing point upon addition of a solute.
\[ \Delta T_f = i K_f m \]
The increase in a solvent’s boiling point upon addition of a solute.
\[ \Delta T_b = i K_b m \]
The pressure required to prevent the inward flow of solvent across a semipermeable membrane.
\[ \Pi = iMRT \]
Thermodynamics & Electrochemistry
Relates heat absorbed or released to mass, specific heat, and temperature change.
\[ q = mc\Delta T \]
Calculates the overall enthalpy change for a reaction from standard molar enthalpies of formation.
\[ \Delta_\text{r} H^\circ = \sum n_{\mathrm{p}}~\Delta_\text{f} H^\circ(\text{products}) - \sum n_{\mathrm{r}}~\Delta_\text{f} H^\circ(\text{reactants}) \]
Calculates the overall entropy change for a reaction from standard molar entropies.
\[ \Delta_\text{r} S^\circ = \sum n_{\mathrm{p}}~S^\circ(\text{products}) - \sum n_{\mathrm{r}}~S^\circ(\text{reactants}) \]
Calculates the standard Gibbs free energy change from standard molar free energies of formation.
\[ \Delta_\text{r} G^\circ = \sum n_{\mathrm{p}}~\Delta_\text{f} G^\circ(\text{products}) - \sum n_{\mathrm{r}}~\Delta_\text{f} G^\circ(\text{reactants}) \]
Relates enthalpy, entropy, and spontaneity at a specified temperature.
\[ \Delta G = \Delta H - T\Delta S \]
Relates entropy to the number of microstates.
\[ S = k_B \ln W \]
Calculates Gibbs free energy under non-standard conditions.
\[ \Delta G = \Delta G^\circ + RT \ln Q \]
Defines entropy change for a reversible process.
\[ \Delta S = \frac{q_\text{rev}}{T} \]
Relates standard Gibbs energy to the equilibrium constant.
\[ \Delta G^\circ = -RT~\ln K \]
Relates Gibbs free energy to electrochemical cell potential.
\[ \Delta G = -nFE_\text{cell} \]
Calculates cell potential under non-standard conditions.
\[ E_\text{cell} = E^\circ_\text{cell} - \frac{RT}{nF}~\ln Q \]
A simplified form of the Nernst equation for standard temperature.
\[ E_\text{cell} = E^\circ_\text{cell} - \frac{0.0257}{n}~\ln Q \]
Relates the standard cell potential to the equilibrium constant.
\[ E^\circ_\text{cell} = \frac{RT}{nF}~\ln K \]
Simplified forms using natural log (ln) and base-10 log.
\[ E^\circ_\text{cell} = \frac{0.0592}{n}~\log K \]
Relates total charge (Q) to current (I), time (t), and moles of electrons (n).
\[ Q = I \times t = n \times F \]
Chemical Kinetics
Relates concentration and time for a zero-order reaction.
\[ [A]_t = -kt + [A]_0 \]
Calculates the half-life of a zero-order reaction.
\[ t_{1/2} = \frac{[\mathrm{A}]_0}{2k} \]
Relates concentration and time for a first-order reaction.
\[ \ln[\mathrm{A}]_t = -kt + \ln[\mathrm{A}]_0 \]
Calculates the half-life of a first-order reaction.
\[ t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k} \]
Relates concentration and time for a second-order reaction.
\[ \frac{1}{[\mathrm{A}]_t} = kt + \frac{1}{[\mathrm{A}]_0} \]
Calculates the half-life of a second-order reaction.
\[ t_{1/2} = \frac{1}{k[\mathrm{A}]_0} \]
Relates the rate constant to activation energy and temperature.
\[ k = Ae^{-E_{\mathrm{a}}/RT} \]
Relates the rate constant to activation energy and temperature.
\[ \ln k = -\dfrac{E_{\mathrm{a}}}{R}\left ( \dfrac{1}{T} \right ) + \ln A \]
A rearrangement of the equation for a line. Calculates activation energy or a rate constant from two data points.
\[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{E_{\mathrm{a}}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]