Topic 6: Gases and Colligative Properties Ideal Gas Equation Boyle noticed an inverse relationship between volume and pressure. Pressure x volume = constant P = a P 1/P Charles found the volume of a gas, at constant pressure, increased linearly with temperature. olume = constant x temperature = bt He Different gases extrapolated to zero volume at the same temperature. This is absolute zero at - 273.15 C = 0 K. CH 4 H 2 Avogadro proposed that equal volumes of gases at the same temperature and pressure contained the same number of particles. Combining these equations gives olume = constant x No of moles = cn T Ideal Gas Equation: P = nrt R is the ideal gas constant (8.314 J K 1 mol 1 or 0.082 L atm K 1 mol 1 ) 52
Example: The mass of 1.00 L of a gas at 2.00 atm and 25 C is 2.76 g. What is the molar mass of the gas? P = nrt and n = m/m (2.0 atm)(1.00 L) = n (0.082 L atm K - 1 mol - 1 )(273 + 25 K) n = (2.00 / 24.436) mol = 0.0818 mol n = mass / molar mass or molar mass = mass/ moles Molar mass = 2.76 g / 0.0818 mol = 33.7 g mol - 1 1. If a gas occupies a volume of 1.5 L at 105 kpa and 20 C, what volume will it occupy at standard temperature and pressure (0 C and 1 atm = 101.3 kpa)? The density of a gas (ρ) depends on its molar mass (M), the pressure (P) and on the temperature (T). For an ideal gas ρ = MP / RT 2. Complete the table below showing the properties of three familiar gases at atmospheric pressure (P = 1.00 atm) and at 273 K. n = 1.00 mol Helium Nitrogen Oxygen olume occupied (in L) Mass (in g) Number of gas particles Density 3. Which of these properties will be different at the same pressure but at 298 K? 4. Which of these properties will be different for a scuba diver at 273 K and 2.00 atm? 5. If air is 79.0% N 2 and 21.0% O 2, what is the density of air at 273 K and 1.00 atm? 53
Partial Pressure In a mixture of gases, the total pressure exerted is the sum of the partial pressures that each gas would exert if it were alone. P TOTAL = p A + p B + p C +. The partial pressure is related to the number of moles present, the mole fraction, x, Mole fraction of A = x A = (No of moles of A) / (Total No of moles present) Thus p A = x A P TOTAL and p B = x B P TOTAL Example: Air consists of approximately 21 % oxygen and 79 % nitrogen. What are the partial pressures of these gases at 7.5 atm? Mole fraction of O 2 = x O2 = (21 %) / (21 % + 79 %) = 0.21 Mole fraction of N 2 = x N2 = (79 %) / (79 % + 21 %) = 0.79 (check: Σ mole fractions = 1) At 10 atmosphere: p O2 = 0.21 x 7.5 atm = 1.6 atm p N2 = 0.79 x 7.5 atm = 5.9 atm (check: Σ partial pressure = total pressure) 6. Exactly 500 ml of a gas contains 0.18 mol of PCl 3 and 0.050 mol of Cl 2 at 298 K. What is the total pressure and partial pressures of these gases? (Hint: i. Calculate total no of moles, ii. Use this and the data given, together with the Ideal Gas Equation, to find P TOTAL iii. Calculate mole fractions, iv. Calculate partial pressures). 54
Kinetic Molecular Theory of Gases The volume of the particle is negligible (zero) The particles are in constant motion No intermolecular forces between particles Average kinetic energy of the gas is directly proportional to the temperature (Kelvin) P = 2/3 { [ n N A ( 1 / 2 m u 2 ) ] / } Relative No with a given speed 273 K 1273 K No of molecules N 2 H 2 O 2273 K He H 2 1000 2000 Speed (m /s) Molecular speed Colligative Properties Boiling- Point Elevation Freezing- Point Depression Osmotic Pressure The presence of a solute affects the physical properties of a solvent. Colligative properties all depend on the number, not the identity, of the solute particles in a solution. Use salt to melt ice on a footpath - causes the freezing point of the water to decrease Add antifreeze to the water in the radiator of a car to lower the freezing point of the water Osmotic Pressure The flow of solvent molecules into a solution through a semi- permeable membrane is called OSMOSIS. In osmosis a semi- permeable membrane prevents the transfer of solute particles. More solvent molecules enter the solution than leave it. As a result the solution volume increases and its concentration decreases. The solution exerts a backward pressure - the osmotic pressure (π) that eventually equalises the flow of solvent molecules in both directions 55
The osmotic pressure is also defined as the applied pressure required to prevent this volume change. π = crt π is the osmotic pressure in atm, c is the total molarity of all solute particles (mol L - 1 ) R is the gas constant and T is the temperature in K Example: A sodium chloride solution has a concentration of 0.15 M. What is the osmotic pressure of this solution at 25 C? π = crt π = (2 x 0.15 mol L - 1 )(0.082 L atm K - 1 mol - 1 )(273 + 25 K) = 7.3 atm (NB The saline solution contains 0.15 M Na + ions and 0.15 M Cl ions, so the total concentration of solute particles is 2 x 0.15 M = 0.30 M). 7. Lysozyme is an enzyme that breaks down bacterial cell walls. A solution containing 0.150 g of this enzyme in 210 ml of solution has an osmotic pressure of 0.00125 atm at 25 C. What is the molar mass of lysozyme? Use π = crt and c = mol / vol and mol = mass/molar mass 56