Equation of state
Section: Ideal Gases | Syllabus: Cambridge A Level Physics (9702)
Boyle's Law
The pressure, p, of a constant mass of gas at constant temperature is inversely proportional to its volume, V.
p ∝ (1/V) or pV = constant Experimental SkillsTo investigate this, air is trapped in a glass tube above oil. As oil is pumped into the tube, the air volume is reduced and the pressure increases. A graph of p versus 1/V yields a straight line through the origin, confirming the relationship.
Charles's Law
The volume, V, of a constant mass of gas at constant pressure is directly proportional to its thermodynamic temperature, T.
V ∝ T or (V/T) = constantThe Pressure Law
For a constant mass of gas at constant volume, the pressure, p, is directly proportional to its thermodynamic temperature, T.
p ∝ T or (p/T) = constant Estimating Absolute ZeroBy plotting a graph of pressure against temperature in ^ and extrapolating the line to zero pressure, the x-intercept identifies Absolute Zero at approximately -273.15 ^.
The Equation of State
Combining the gas laws for an Ideal Gas (a theoretical gas that obeys these laws under all conditions):
pV = nRTWhere n is the number of moles and R is the molar gas constant (R = 8.31 J K^-1 mol^-1).
pV = NkTWhere N is the number of molecules and k is the Boltzmann constant (k = (R/N_A) = 1.38 × 10^-23 J K^-1).
Worked Examples
Worked Example: Ideal Gas CalculationQuestion: A gas in a cylinder has a volume 5.2 × 10^-2 m^3 at 22 ^ and 520 kPa. Determine the number of molecules and the mass of the gas if its relative molecular mass is 32.
Solution1. T = 22 + 273.15 = 295.15 K.
2. n = pV / RT = (520 × 10^3 × 5.2 × 10^-2) / (8.31 × 295.15) = 11.02 mol.
3. N = n × N_A = 11.02 × 6.02 × 10^23 = 6.6 × 10^24 molecules.
4. Mass = 11.02 × 32 = 352.6 g = 0.35 kg.
Figure 15.3: Boyle's Law ApparatusSchematic showing a glass tube with trapped air above oil, connected to a Bourdon pressure gauge and a pump. The volume scale allows reading air volume as pressure is varied.
Figure 15.4: Gas Law GraphsA composite diagram showing:
a) Boyle's Law: p vs V (curve) and p vs 1/V (straight line).
b) Charles's Law: V vs T(K) (straight line through origin).
c) Pressure Law: p vs T(K) (straight line through origin) and p vs T(^) (extrapolated to -273.15 ^).
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