Essential Gas Laws and Ideal Gas Equations

1. Gas Laws

All gas samples can be described by 4 properties (m, n, V, T). These properties are easily measured in gases. All gases behave similarly when varying P, V, and T.

1.1. PV Relationship (Boyle's Law)

When T is constant: P_i · V_i = P_f · V_f (1 atm = 1013 × 10⁵ Pa; 1 atm = 760 mmHg).

1.2. PT Relationship (Gay-Lussac's Law)

When V is constant: P_i / T_i = P_f / T_f.

1.3. PVT Relationship (Combined Gas Law)

For any mass of gas, the product of P and V divided by T is constant: (P_i · V_i) / T_i = (P_f · V_f) / T_f.

1.4. Avogadro's Hypothesis

If different gases are at the same P, T, and V, they contain the same number of molecules. Therefore, 1 mole of gas occupies the same volume under the same conditions. (Standard conditions: 1 atm, 273 K, 22.4 L).

1.5. Ideal Gas Law

PV = nRT

2. Molecular Mass of a Gas

From the ideal gas formula, we can calculate the molar mass (M): M = (m · R · T) / (P · V), where m is in kg, R = 8.31, P in Pa, V in m³, and T in K.

• Empirical formula: Divide the number of moles of each substance by the smallest number of moles.
• Molecular formula: Divide the calculated molar mass by the empirical formula mass to find the multiplier for the empirical formula.

3. Gas Density

There are two ways to calculate gas density:

1. Mass and volume: d = m / V
2. Ideal gas relation: d = (P · M) / (R · T)

Density increases when T decreases or M increases.

4. Gas Mixtures

Gas laws apply to mixtures where all gases are at the same T and distributed uniformly throughout volume V. Each gas exerts a partial pressure proportional to its molar fraction. The total pressure is the sum of partial pressures.

For a single gas: P_gas = (n_gas · R · T) / V
For a mixture: P_total = (n_total · R · T) / V
Molar fraction: X_gas = n_gas / n_total