Mathematical derivation of Dalton's law and their applications

  • Partial pressure of a gas in a mixture of gases is the pressure which that gas would exert if it were the only gas present in the container.

  • Dalton's Law of Partial Pressures states that the total pressure in a gas mixture is the sum of the partial pressures of each individual gas.

    Ptotal = Pgas a + Pgas b + Pgas c + etc

  • Dalton's Law of Partial Pressures assumes each gas in the mixture is behaving like anideal gas.

Partial Pressure

A container of fixed volume at constant temperature holds a mixture of gas a and gas b at a total pressure of 4atm.

The total pressure in the container is proportional to the number of gas particles.

More gas particles = greater pressure.
Less gas particles = lower pressure.

If each dot represents 1 mole of gas particles, then there are 48 moles of gas particles in this container exerting a total pressure of 4atm.

Imagine the container with no particles of gas b.

Only particles of gas a are present in the same container at the same temperature.

Now the container holds only 12 moles of gas particles instead of the 48 moles of gas particles it originally contained.

Since pressure is proportional to the number of gas particles,
the pressure exerted by gas a = 12mol ÷ 48mol x 4atm = 1atm

Imagine the container with no particles of gas a.

Only particles of gas b are present in the same container at the same temperature.

Now the container holds only 36 moles of gas particles instead of the 48 moles of gas particles it originally contained.

Since pressure is proportional to the number of gas particles,
the pressure exerted by gas b = 36mol ÷ 48mol x 4atm = 3atm

Dalton's Law of Partial Pressures

The total pressure in a gas mixture is the sum of the partial pressures of each individual gas.

Ptotal=Pgas a+Pgas b
=+

Examples

  1. 10g of nitrogen gas and 10g of helium gas are placed together in a 10L container at 25oC. Calculate the partial pressure of each gas and the total pressure of the gas mixture.

    Calculate the moles (n) of each gas present: n = mass ÷ molecular mass

    nitrogen (N2(g))helium (He(g))
    mass (g)10g10g
    molecular mass (g/mol)2 x 14 = 284
    n = mass ÷ molecular mass10 ÷ 28 = 0.4mol10 ÷ 4 = 2.5mol

    Calculate the total moles of gas present = 0.4 + 2.5 = 2.9mol

    Calculate the total gas pressure assuming ideal gas behaviour: PV = nRT
    P = n x R x T ÷ V
    n = 2.9mol
    R = 8.314
    T = 25    oC = 25 + 273 = 298K
    V = 10L
    P = 2.9 x 8.314 x 298 ÷ 10 = 718kPa (7atm)

    Partial pressure of nitrogen = n(N2) ÷ n(total) x total pressure
    Partial pressure of nitrogen = 0.4 ÷ 2.9 x 718kpa = 99kPa (0.9atm)

    Partial pressure of helium = n(He) ÷ n(total) x total pressure
    Partial pressure of helium = 2.5 ÷ 2.9 x 718 = 619kPa (6.1atm)

  2. At 15oC, 25mL of neon at 101.3kPa (1atm) pressure and 75mL of helium at 70.9kPa (0.7atm) pressure are both expanded into a 1L sealed flask. Calculate the partial pressure of each gas and the total pressure of the gas mixture.

    Since the temperature and moles of each gas is constant, the pressure exerted by each gas is inversely proportional to its volume (Boyle's Law).
    P    iVi = PfVf
    P    f = PiVi ÷ Vf

    Partial pressure Neon = 101.3kPa x 25 x 10-3L ÷ 1L = 2.5kPa

    Partial pressure Helium = 70.9kPa x 75 x 10-3L ÷ 1L = 5.3kPa

    Total pressure = 2.5 + 5.3 = 7.8kPa

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