8. Gases
Gases are the most predictable state of matter: pressure, volume, temperature, and moles are linked by a handful of laws that all collapse into one equation.
The Ideal Gas Law
$$PV = nRT, \qquad R = 0.08206\ \text{L·atm/mol·K}$$
Example: Moles in 2.0 L of gas at 1.0 atm and 273 K:
$$n = \frac{PV}{RT} = \frac{(1.0)(2.0)}{(0.08206)(273)} = 0.089\ \text{mol}$$
At STP (0 °C, 1 atm), one mole occupies 22.4 L.
Combined Gas Law and Dalton's Law
$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \qquad P_{\text{total}} = P_1 + P_2 + \cdots$$ Partial pressure follows mole fraction: $P_i = X_i P_{\text{total}}$.
Kinetic Molecular Theory and Effusion
Gas particles move randomly, collide elastically, and have average kinetic energy $\tfrac{3}{2}RT$. Graham's law: lighter gases effuse faster: $$\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$$
Example: $\text{H}_2$ effuses $\sqrt{32/2} = 4$ times faster than $\text{O}_2$ at the same temperature.
Practice Problems
- What volume does 0.50 mol of gas occupy at STP?
- Find the pressure of 2.0 mol of gas in a 5.0 L container at 300 K.
- Which effuses faster, He or Ar, and by what factor?
- A mixture at 2.0 atm total pressure contains 0.30 mol O2 out of 1.0 mol total gas. What is the partial pressure of O2?
Answers: 1. 11.2 L 2. 9.85 atm 3. He, by a factor of √(40/4) = 3.16 4. 0.60 atm