Solution & Colligative Property Equations and Reaction Kinetics

Solution & Colligative Property Equations

Henry's Law

Henry's Law describes the relationship between the solubility of a gas in a liquid and the partial pressure of the gas above the liquid.

• Equation: $S_{gas} = k_H P_{gas}$

• Key Terms: $S_{gas}$ = solubility of the gas, $k_H$ = Henry's law constant, $P_{gas}$ = partial pressure of the gas

• Application: Used to calculate how much gas will dissolve in a liquid under a given pressure.

Raoult's Law

Raoult's Law relates the vapor pressure of a solution to the mole fraction of the solvent and its pure vapor pressure.

• Equation: $P_{solution} = X_{solvent} P^0_{solvent}$

• Key Terms: $P_{solution}$ = vapor pressure of the solution, $X_{solvent}$ = mole fraction of the solvent, $P^0_{solvent}$ = vapor pressure of pure solvent

• Application: Used to predict how the addition of a solute affects the vapor pressure of a solvent.

Freezing Point Depression & Boiling Point Elevation

Colligative properties depend on the number of solute particles in solution, not their identity.

• Freezing Point Depression: $\Delta T_f = i m K_f$

• Boiling Point Elevation: $\Delta T_b = i m K_b$

• Key Terms: $i$ = van't Hoff factor, $m$ = molality, $K_f$ = freezing point depression constant, $K_b$ = boiling point elevation constant

• Application: Used to calculate changes in freezing and boiling points when a solute is added.

Osmotic Pressure

Osmotic pressure is the pressure required to stop the flow of solvent through a semipermeable membrane separating solutions of different concentrations.

• Equation: $\Pi = iMRT$

• Alternate Forms: $\Pi = iCRT$

• Key Terms: $\Pi$ = osmotic pressure, $i$ = van't Hoff factor, $M$ = molarity, $C$ = concentration, $R$ = gas constant, $T$ = temperature (K)

• Application: Used to determine the osmotic pressure of solutions, important in biological and chemical systems.

Chemical Kinetics: Reaction Rate Equations

Zero Order Reaction

Zero order reactions have a constant rate independent of the concentration of reactants.

• Integrated Rate Law: $[A] = -kt + [A]_0$

• Half-life: $t_{1/2} = \frac{[A]_0}{2k}$

• Application: Common in reactions where a catalyst is saturated by reactant.

First Order Reaction

First order reactions have a rate proportional to the concentration of one reactant.

• Integrated Rate Law: $\ln [A] = -kt + \ln [A]_0$

• Alternate Form: $[A] = [A]_0 e^{-kt}$

• Half-life: $t_{1/2} = \frac{0.693}{k}$

• Application: Radioactive decay and many simple chemical reactions follow first order kinetics.

Second Order Reaction

Second order reactions have a rate proportional to the square of the concentration of one reactant or the product of two reactant concentrations.

• Integrated Rate Law: $\frac{1}{[A]} = kt + \frac{1}{[A]_0}$

• Half-life: $t_{1/2} = \frac{1}{k[A]_0}$

• Application: Many bimolecular reactions are second order.

Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to temperature and activation energy.

• Equation: $k = Ae^{-E_a/(RT)}$

• Linear Form: $\ln k = -\frac{E_a}{R} \left( \frac{1}{T} \right) + \ln A$

• Two-Point Form: $\ln \left( \frac{k_2}{k_1} \right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

• Key Terms: $k$ = rate constant, $A$ = frequency factor, $E_a$ = activation energy, $R$ = gas constant ($8.314$ J/mol·K), $T$ = temperature (K)

• Application: Used to analyze how reaction rates change with temperature.

Summary Table: Kinetics Equations

The following table summarizes the main equations for zero, first, and second order reactions, as well as the Arrhenius equation.

Order	Integrated Rate Law	Half-life Equation
Zero Order	$[A] = -kt + [A]_0$	$t_{1/2} = \frac{[A]_0}{2k}$
First Order	$\ln [A] = -kt + \ln [A]_0$ or $[A] = [A]_0 e^{-kt}$	$t_{1/2} = \frac{0.693}{k}$
Second Order	$\frac{1}{[A]} = kt + \frac{1}{[A]_0}$	$t_{1/2} = \frac{1}{k[A]_0}$

Additional info: These equations are essential for understanding solution properties and chemical kinetics, which are core topics in General Chemistry (Ch. 14, Ch. 15, and related chapters).