Delta T_f = i K_f m

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

Freezing-point depression. Use it when molality m.

Delta T_f = i K_f m

pi = i C R T

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

Osmotic pressure. Use it when molar concentration C.

pi = i C R T

p_solution = x_solvent * p_pure

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

Raoult's law. Use it when ideal solution.

p_solution = x_solvent * p_pure

p_gas = K_H * x_gas

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

Henry's law. Use it when gas dissolved in a liquid.

p_gas = K_H * x_gas

boiling-point elevation vs freezing-point depression

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

Adding solute raises the boiling point and lowers the freezing point; both scale with particle count and molality.

ideal solution vs non-ideal solution

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

An ideal solution obeys Raoult's law with no enthalpy or volume change; non-ideal solutions deviate and can form azeotropes.

osmosis vs diffusion

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

Diffusion is any particle spreading down a gradient; osmosis is specifically solvent moving across a semi-permeable membrane toward higher solute concentration.

Watch out: Colligative properties depend on the type of solute

“Antifreeze and salted roads both exploit colligative lowering of the freezing point.”

They depend only on the NUMBER of dissolved particles, which is why electrolytes need the van't Hoff factor.

Unit cells and packing

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Crystals repeat a unit cell; fcc and hcp pack most efficiently at about 74%.

Memory trick: fcc/hcp = 74%, bcc = 68%, simple cubic = 52%.

Unit cells and packing — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is thinking simple cubic packs efficiently. In reality, crystals repeat a unit cell; fcc and hcp pack most efficiently at about 74%.

Memory trick: fcc/hcp = 74%, bcc = 68%, simple cubic = 52%.

Atoms per unit cell

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Counting shared atoms gives 1 for simple cubic, 2 for bcc and 4 for fcc.

Memory trick: corner = 1/8, face = 1/2, body = 1.

Atoms per unit cell — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is counting corner atoms as whole atoms. In reality, counting shared atoms gives 1 for simple cubic, 2 for bcc and 4 for fcc.

Memory trick: corner = 1/8, face = 1/2, body = 1.

Coordination number

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

The coordination number rises with packing: 6 (simple cubic), 8 (bcc), 12 (fcc/hcp).

Memory trick: denser packing -> higher coordination number.

Coordination number — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is assuming all cubic lattices have the same coordination. In reality, the coordination number rises with packing: 6 (simple cubic), 8 (bcc), 12 (fcc/hcp).

Memory trick: denser packing -> higher coordination number.

Voids

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Close packing leaves tetrahedral and octahedral holes that smaller ions occupy in ionic solids.

Memory trick: N atoms give N octahedral and 2N tetrahedral voids.

Voids — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is mixing up the number of each void type. In reality, close packing leaves tetrahedral and octahedral holes that smaller ions occupy in ionic solids.

Memory trick: N atoms give N octahedral and 2N tetrahedral voids.

Point defects

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Schottky defects lower density (missing ion pairs); Frenkel defects keep density (ion shifts to an interstitial).

Memory trick: Schottky lowers density, Frenkel keeps it.

Point defects — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is swapping which defect changes the density. In reality, Schottky defects lower density (missing ion pairs); Frenkel defects keep density (ion shifts to an interstitial).

Memory trick: Schottky lowers density, Frenkel keeps it.

Semiconductors and doping

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Doping silicon with group 15 gives n-type, group 13 gives p-type.

Memory trick: extra electrons (group 15) = n-type.

Semiconductors and doping — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is reversing which group gives n- or p-type. In reality, doping silicon with group 15 gives n-type, group 13 gives p-type.

Memory trick: extra electrons (group 15) = n-type.

Magnetic and dielectric solids

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Ferromagnetic, antiferromagnetic and ferrimagnetic order differ in how spins align.

Memory trick: parallel spins = ferro; opposed and unequal = ferri.

Magnetic and dielectric solids — common mistake

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

A frequent error is treating all magnetic solids as ferromagnetic. In reality, ferromagnetic, antiferromagnetic and ferrimagnetic order differ in how spins align.

Memory trick: parallel spins = ferro; opposed and unequal = ferri.

Packing efficiency: fcc 74%, bcc 68%, simple cubic 52%

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Fraction of space filled. Use it when ideal hard spheres.

Packing efficiency: fcc 74%, bcc 68%, simple cubic 52%

Z = 1 (sc), 2 (bcc), 4 (fcc)

“Doping pure silicon with a pinch of impurity turns it into the semiconductor behind all electronics.”

Atoms per unit cell. Use it when counting shared atoms.

Z = 1 (sc), 2 (bcc), 4 (fcc)