AP Chemistry · Lesson 2

Ideal and real gases

Know Know it

The ideal gas law is PV=nRTPV = nRT. Use R=0.08206R = 0.08206 L·atm/(mol·K) with pressure in atm and volume in L, and always use kelvin: K = °C + 273.15.

For one sample whose moles stay the same, P1V1T1=P2V2T2\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}. Rearranging PV=nRTPV = nRT also gives the molar mass, M=mRTPVM = \frac{mRT}{PV}, and the density, d=PMRTd = \frac{PM}{RT}.

Dalton's law: in a mixture, Ptotal=PA+PB+…P_{total} = P_A + P_B + \ldots, and each partial pressure is PA=XAPtotalP_A = X_A P_{total}, where XAX_A is the mole fraction of A.

Kinetic molecular theory: the average kinetic energy of the particles depends only on the kelvin temperature. At the same temperature, lighter particles move faster. A Maxwell-Boltzmann curve for a hotter or lighter gas is flatter and peaks at a higher speed.

Real gases behave least ideally at high pressure (the particles' own volume is no longer negligible) and low temperature (attractions pull particles together and lower the pressure). Polar or large gases depart from ideal more than small nonpolar ones.

Calculator On the exam

A scientific or graphing calculator is allowed on both sections of the exam, and a four-function one is permitted but not recommended. The equations and constants sheet gives R and other constants. Write the equation and each value with its unit before you calculate, and round only at the end.

Worked example Easy

A rigid 5.00 L tank holds 0.250 mol of helium at 25 °C. What is the pressure in the tank? (R = 0.08206 L·atm/(mol·K))

  1. 0.103 atm
  2. 0.818 atm
  3. 1.22 atmAnswer
  4. 124 atm

How to solve it

  1. Convert temperature: 25 + 273 = 298 K.
  2. P=nRTV=(0.250)(0.08206)(298)5.00P = \frac{nRT}{V} = \frac{(0.250)(0.08206)(298)}{5.00}.
  3. P=1.22P = 1.22 atm, to three significant figures.

Why each choice is right or wrong

  • A. Incorrect. 0.103 atm uses 25 instead of 298 K; the temperature in the gas law must be in kelvin.
  • B. Incorrect. 0.818 atm is VnRT\frac{V}{nRT}, the reciprocal of the pressure, so it inverts the rearranged gas law.
  • C. Correct. P=nRTV=0.250×0.08206×2985.00=1.22P = \frac{nRT}{V} = \frac{0.250 \times 0.08206 \times 298}{5.00} = 1.22 atm.
  • D. Incorrect. 124 atm uses R = 8.314 J/(mol·K), which gives a pressure in kPa, not in atm.

Trap Kelvin, always

Doubling a temperature in °C does not double the kinetic energy. Going from 27 °C to 127 °C is going from 300 K to 400 K, so the volume at constant pressure grows by a factor of 400/300, not 127/27. Convert every temperature to kelvin before you use a gas law.

Explain Explain it

Task verbs: calculate, explain

To find a molar mass from a gas sample, find moles from PV = nRT, then divide the mass by those moles. Check the units: g ÷ mol = g/mol.

For a mixture, the mole fraction of a gas is also its share of the total pressure, because every gas in the container has the same volume and temperature.

Worked example Medium

A 1.20 g sample of an unknown gas fills a 0.500 L flask at 1.00 atm and 300. K. What is the molar mass of the gas? (R = 0.08206 L·atm/(mol·K))

  1. 0.0169 g/mol
  2. 53.8 g/mol
  3. 59.1 g/molAnswer
  4. 0.0244 g/mol

How to solve it

  1. Moles: n=PVRT=(1.00)(0.500)(0.08206)(300.)=0.0203n = \frac{PV}{RT} = \frac{(1.00)(0.500)}{(0.08206)(300.)} = 0.0203 mol.
  2. Molar mass: M=mn=1.200.0203=59.1M = \frac{m}{n} = \frac{1.20}{0.0203} = 59.1 g/mol.

Why each choice is right or wrong

  • A. Incorrect. 0.0169 g/mol divides the moles by the mass, which gives mol/g, the reciprocal of a molar mass.
  • B. Incorrect. 53.8 g/mol uses 273 K instead of the 300. K given, which gives too many moles and too small a molar mass.
  • C. Correct. n=PVRT=(1.00)(0.500)(0.08206)(300.)=0.0203n = \frac{PV}{RT} = \frac{(1.00)(0.500)}{(0.08206)(300.)} = 0.0203 mol, and 1.20 g0.0203 mol=59.1\frac{1.20 \text{ g}}{0.0203 \text{ mol}} = 59.1 g/mol.
  • D. Incorrect. 0.0244 g/mol multiplies the mass by the moles instead of dividing, which gives the wrong units for a molar mass.

Calculate Your turn

A sealed container holds 3.00 mol of nitrogen gas and 1.00 mol of oxygen gas at a total pressure of 2.00 atm. Calculate the partial pressure of oxygen. Then explain why the two gases have the same average kinetic energy even though their molecules have different masses.

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Compare with a model answer

A full answer does this

  • Finds the mole fraction of oxygen, 1.00 ÷ 4.00 = 0.250, and the partial pressure 0.500 atm
  • Explains that average kinetic energy depends only on temperature, and the two gases are at the same temperature

Model answer

The total is 3.00 + 1.00 = 4.00 mol, so the mole fraction of oxygen is 1.00 ÷ 4.00 = 0.250, and its partial pressure is 0.250 × 2.00 atm = 0.500 atm. By kinetic molecular theory, the average kinetic energy of gas particles depends only on the kelvin temperature. Both gases are in the same container at the same temperature, so their average kinetic energies are equal; the lighter N₂ molecules simply move a little faster on average.

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