Gas Laws and the Ideal Gas Equation (PV = nRT) - Chemistry

1. Fundamental Concepts

Gas Properties: Gases have no fixed volume or shape, exhibit high compressibility, and their particles move randomly with negligible intermolecular forces (ideal gas assumption).
State Variables: Physical quantities defining a gas state:
- : Pressure (units: atm, kPa, mmHg)
- : Volume (units: L, m³)
- : Absolute Temperature (units: K; )
- : Moles of gas (units: mol)
Ideal Gas Assumption: No molecular volume, no intermolecular forces, elastic collisions between particles and container walls.

2. Key Concepts

2.1 Simple Gas Laws (Special Cases of PV = nRT)

Law	Condition	Relationship
Boyle’s Law	Constant ,	(inverse)
Charles’s Law	Constant ,	(direct)
Gay-Lussac’s Law	Constant ,	(direct)
Avogadro’s Law	Constant ,	(direct)

2.2 Ideal Gas Equation

Formula:
Gas Constant (): Depends on units of P and V:
- (most common for lab units)
- (SI units)
Combined Gas Law: For changing state variables (constant ):
$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

3. Examples

Easy(Boyle’s Law Application)

Problem: A gas occupies 5.0 L at 2.0 atm. What is its volume at 4.0 atm (constant $n$, $T$)?

Solution: Use $P_1V_1 = P_2V_2$

$V_2 = \frac{P_1V_1}{P_2} = \frac{2.0\ \text{atm} \times 5.0\ \text{L}}{4.0\ \text{atm}} = 2.5\ \text{L}$

Medium (Ideal Gas Equation Application)

Problem: Calculate the pressure of 0.50 mol of $O_2$ in a 10.0 L container at 25°C.

Solution:

1. Convert $T$ to K: $25 + 273.15 = 298.15\ \text{K}$

2. Use $PV = nRT$ with $R = 0.0821\ \text{L·atm/(mol·K)}$

$P = \frac{nRT}{V} = \frac{0.50\ \text{mol} \times 0.0821\ \text{L·atm/(mol·K)} \times 298.15\ \text{K}}{10.0\ \text{L}} \approx 1.22\ \text{atm}$

Hard (Combined Gas Law + Molar Mass)

Problem: A 2.00 g sample of an unknown gas occupies 1.50 L at 1.00 atm and 27°C. Find its molar mass ($M$).

Solution:

1. Convert $T$ to K: $27 + 273.15 = 300.15\ \text{K}$

2. Relate moles to molar mass: $n = \frac{m}{M}$

3. Substitute into $PV = nRT$: $PV = \frac{mRT}{M}$

4. Solve for $M$:

$M = \frac{mRT}{PV} = \frac{2.00\ \text{g} \times 0.0821\ \text{L·atm/(mol·K)} \times 300.15\ \text{K}}{1.00\ \text{atm} \times 1.50\ \text{L}} \approx 32.9\ \text{g/mol}$

4. Problem-Solving Techniques

Unit Consistency: Ensure units match , , units; always use Kelvin for temperature.
Identify Constant Variables: Determine which state variables are fixed to select the correct gas law (simple law vs. ideal gas equation vs. combined law).
Mole Conversion: Use $n = \frac{m}{M}$ (mass to moles) or $n = \frac{V_{\text{STP}}}{22.4\ \text{L/mol}}$ (volume at STP to moles; STP = 1 atm, 273 K) when needed.
Stepwise Calculation: For multi-step problems (e.g., molar mass, density), break down into smaller steps (convert units → select formula → solve for unknown).
Check Assumptions: Verify if the gas behaves ideally (high temperature, low pressure); real gas corrections are not required for basic problems.