Key Equations and Concepts in General Chemistry and Thermodynamics

Kinetics and Thermodynamics

Rate Laws and Integrated Rate Equations

Understanding how chemical reactions proceed over time is fundamental in chemistry. The rate of a reaction can be described mathematically using rate laws and integrated rate equations.

• First-Order Reactions: The rate depends linearly on the concentration of one reactant.

• Integrated Rate Law (First Order):

$\ln \left( \frac{[A]_t}{[A]_0} \right) = -kt$

• Second-Order Reactions: The rate depends on the concentration of two reactants or the square of one reactant.

• Integrated Rate Law (Second Order):

$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

• Zero-Order Reactions: The rate is independent of the concentration of reactants.

• Integrated Rate Law (Zero Order):

$[A]_t = [A]_0 - kt$

• Arrhenius Equation: Describes the temperature dependence of reaction rates.

$k = A e^{-E_a/(RT)}$

$\ln k = \ln A - \frac{E_a}{RT}$

$\ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$

• Activation Energy ($E_a$): The minimum energy required for a reaction to occur.

Thermodynamics

• Gibbs Free Energy ($\Delta G$): Determines the spontaneity of a reaction.

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

• Standard Free Energy Change:

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

• Relationship to Equilibrium Constant:

$\Delta G^\circ = -RT \ln K$

• Non-Standard Conditions:

$\Delta G = \Delta G^\circ + RT \ln Q$

• Van't Hoff Equation: Relates the change in equilibrium constant with temperature.

$\ln \left( \frac{K_2}{K_1} \right) = \frac{\Delta H^\circ}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$

Equilibrium Theory

Acid-Base Equilibria

• Henderson-Hasselbalch Equation: Used to calculate the pH of buffer solutions.

$pH = pK_a + \log \left( \frac{[A^-]}{[HA]} \right)$

• Relationship between pH and pOH:

$pH + pOH = 14$

Equilibrium Constants

• General Equilibrium Constant:

$K_p = K_c (RT)^{\Delta n}$

• Partial Pressure and Mole Fraction:

$P_A = X_A \cdot P_{A}^\ast$

Electrochemistry

Cell Potentials and the Nernst Equation

• Standard Cell Potential: The voltage of an electrochemical cell under standard conditions.

• Nernst Equation: Relates cell potential to concentrations of reactants and products.

$E_{cell} = E_{cell}^\circ - \frac{0.0592}{n} \log Q$ at $25^\circ C$

• Gibbs Free Energy and Cell Potential:

$\Delta G^\circ = -nFE^\circ_{cell}$

Gas Laws

Ideal Gas Law

• Equation of State for Ideal Gases:

$PV = nRT$

• Osmotic Pressure:

$\Pi = MRT$

Constants and Useful Values

• Gas Constant (R):

$R = 8.314 \ \frac{J}{mol \cdot K}$ $R = 0.0821 \ \frac{L \cdot atm}{mol \cdot K}$

• Faraday's Constant (F):

$F = 96,500 \ \frac{J}{mol \cdot V}$

Mathematical Tools

• Quadratic Formula: Used to solve quadratic equations, often encountered in equilibrium calculations.

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Summary Table: Key Equations and Constants

Equation/Constant	Description
$PV = nRT$	Ideal Gas Law
$\Delta G = \Delta H - T\Delta S$	Gibbs Free Energy
$E_{cell} = E_{cell}^\circ - \frac{0.0592}{n} \log Q$	Nernst Equation at 25°C
$pH = pK_a + \log \left( \frac{[A^-]}{[HA]} \right)$	Henderson-Hasselbalch Equation
$K_p = K_c (RT)^{\Delta n}$	Relationship between $K_p$ and $K_c$
$R = 8.314 \ \frac{J}{mol \cdot K}$	Gas Constant (SI units)
$F = 96,500 \ \frac{J}{mol \cdot V}$	Faraday's Constant

Additional info: Some equations and constants are more commonly used in general chemistry, but are also foundational for understanding biological processes such as cellular respiration, enzyme kinetics, and membrane transport in General Biology.