Gases and the gas laws
Gases expand to fill containers and respond predictably to pressure and temperature. A few simple laws describe them remarkably well.
Describing a gas
A gas sample is described by pressure \(P\), volume \(V\), temperature \(T\) (in kelvin), and amount \(n\) (moles). Pressure comes from molecules colliding with the container walls. Standard atmospheric pressure is 1 atm \(= 101.325\) kPa \(= 760\) mmHg.
Boyle's and Charles's laws
At constant temperature, pressure and volume are inversely proportional: \(P_1V_1 = P_2V_2\) (Boyle). Squeezing a syringe raises the pressure. At constant pressure, volume is proportional to absolute temperature: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) (Charles). A balloon shrinks in the freezer.
Temperature must be in kelvin
Gas laws use absolute temperature: \(T(\text{K}) = T(^\circ\text{C}) + 273.15\). At \(0\) K (absolute zero), an ideal gas would have zero volume. Using Celsius in gas law calculations gives nonsense, such as negative volumes.
The ideal gas law
Combining the laws gives \(PV = nRT\), with \(R = 0.08206\) L atm/(mol K) or 8.314 J/(mol K). At standard temperature and pressure (0 \(^\circ\)C, 1 atm), one mole of ideal gas occupies about 22.4 L.
PV = nRTWorked example
What volume does 2.00 mol of gas occupy at 300 K and 1.50 atm? \(V = \frac{nRT}{P} = \frac{2.00(0.08206)(300)}{1.50} \approx 32.8\) L.
Dalton's law of partial pressures
In a mixture, each gas contributes its own partial pressure, and the total is their sum: \(P_{\text{total}} = P_1 + P_2 + \cdots\). Air at sea level is about 21% oxygen, so oxygen's partial pressure is about 0.21 atm. At high altitude the total pressure drops, lowering the oxygen available to climbers.
Kinetic molecular theory
The gas laws follow from a model: gas molecules are tiny, move randomly, collide elastically, and have average kinetic energy proportional to absolute temperature. Lighter molecules move faster at the same temperature, which is why helium escapes balloons quickly (Graham's law of effusion).
Real gases
Real gases deviate from ideal behaviour at high pressure (molecules' own volume matters) and low temperature (attractions matter). The van der Waals equation corrects for both. Under ordinary conditions, the ideal gas law is accurate to within a few percent.
Key takeaways
Gases are described by \(P\), \(V\), \(T\), and \(n\). Boyle's law: \(PV\) constant; Charles's law: \(V/T\) constant. Always use kelvin. \(PV = nRT\) unifies them; 1 mol occupies 22.4 L at STP. Partial pressures add, and kinetic molecular theory explains the laws.
Flashcard
Force per area from molecular collisions
Check your understanding
A gas at 2.0 atm occupies 6.0 L. At constant temperature, what volume does it occupy at 3.0 atm?