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Gas mixtures

This video covers how we can adapt the ideal gas law for mixtures of gases, and introduces partial pressures.
Now we’ve covered ideal gases and some of the  applications, we’re going to expand into mixtures   of gases. Much of this is the same mathematics  and theory, but just two gases happening at once.  So, we should be happy with properties of gases.  Things like pressure and temperature. And,   of course, the ideal gas law. So now we’re going  to move onto mixtures, which requires looking   at partial pressures and mole fractions.
Partial pressure and mole fractions are   really useful concepts to get, because  they’re analogous to concentration,   which we’re used to in solution chemistry.  So most equations that require concentration,   can apply to gases simply by swapping out  concentration from fraction or partial pressure.  So… let’s look at partial pressure.  If we have a container with a gas, let’s call  it Gas A, because it doesn’t matter what it is,   really. And then we pressurise that container  with a different gas, again we don’t care so   let’s just call it Gas B. Then the pressure  goes up, obviously, as there’s more material   in there. But, some of that pressure comes from  A and the rest from B. We call those individual   pressures a partial pressure.
It happens to  be the pressure the system would be at if we   had the same volume, temperature and number  of moles, but only of one of those gases.  Everything that’s true about pressure  is also true about partial pressure.  So, we can create pV=nRT for two gases,  and rearrange slightly to show we have two   partial pressures, labelled with subscripts. The  number of moles will be different, but otherwise   the other values, temperature and volume, would  be the same. If we removed Gas A and allowed Gas   B to expand to that volume, the total pressure  would be the same as the partial pressure.  The sum of the partial pressures is  the total pressure. So, we can add   those.
And we can also simplify it a little  because most of those are the same value.  This isn’t anything terribly new, it’s just  telling us that pressure relates to the total   number of moles of gas in the system. And  if there are two different gases in there,   then the total number of moles is the  sum of the moles of those two gases.  With that in mind, let’s look at mole fractions A mole fraction, usually represented by the letter   x, is simply a fraction of the total number of  moles.
The moles of the first gas, divided by the   total number of moles – which in the  case is simply the sum of both gases.  But, with a simple bit of rearrangement we can  begin to put that into the ideal gas law again,   substituting that total number of moles for  the moles of one and its inverse mole fraction.  Okay, so where are we going with this…? This  is the point, if we rearrange that again,   we get partial pressure. So, the  total pressure, multiplied by the mole   fraction, is equal to the partial pressure. This is telling us that mole fractions,   and partial pressures are directly proportional.  Which should be obvious, but it’s nice to   see the mathematics tell us the same thing.
It’s this fact that allows us to use partial   pressures as completely synonymous  with concentration when it comes to   equilibrium constants. An equilibrium  between a and b, when these are gases,   is expressed using partial pressures. So, the equilibrium for, say, the formation of   ammonia, should be expressed in partial pressures.  Here, the subscripts under the p giving us the   identity of the gas, rather than being put  in square brackets as we do with solutions.  Now, just to finish off, we’re going  to cover ideal conditions . These are   the conditions where all of the formulas  and equations we’ve seen actually work.  Ideal conditions assume no interactions. We  literally assume the molecules never touch   each other.
Any collisions that might happen have  no real effect on anything. So, naturally, that’s   valid at low concentrations. If the concentration  is low, then the odds of collisions are low.  Ideal conditions are also more valid at higher  temperatures. We’ll get onto this when we cover   non-ideal gases later, but for now high  temperatures give the gases more energy,   and override any attractive forces they may have,  which is what causes deviations from ideality.  Non-ideal gas pressure is also known as fugacity.  This is related to partial pressure by a simple   coefficient that has temperature  and pressure dependence.  This is just a small glimpse  into where we’re going   next with thermodynamic gases.
As we begin to  compress real gases, they begin to interact,   and so we need to start modifying these  equations to take that into account.

We don’t always work with a pure gas. Gases are often mixtures. For instance, the atmosphere is a mixture of nitrogen and oxygen.

Gas mixtures

The ideal gas law applies to both a total number of moles of gas, and also any subset of gases – i.e., different components in a mixture. The pressure of each component gas is referred to as “partial pressure”. This is the fraction of a pressure that is due to that particular gas.

Gas Equation
Gas 1 (p_1=frac{n_1RT}{V})
Gas 2 (p_2=frac{n_2RT}{V})
Both (p_{tot}=p_1+p_2=frac{left(n_1+n_2right)RT}{V})

Partial pressures are important because they are analogous to concentration. Everything involving equilibria and energy, that uses concentrations, can apply to gas phase reactions using partial pressures.

Mole fractions

A related concept is the mole fraction. This is the fraction (in terms of moles) of a gas in the mixture. In a system with two gases, 1 and 2, the mole fraction is:

$$ x_1=frac{n_1}{n_1+n_2} $$

In a mixture of 0.25 moles hydrogen and 0.5 mol of oxygen gases, the molar fraction of hydrogen is:

[frac{0.25}{0.25 + 0.5} = frac{0.25}{0.75} = 0.33]

A partial pressure is the mole fraction of the gas multiplied by the total pressure. So if the above hydrogen and oxygen mixture was at a pressure of 1.5 atm, the partial pressure of hydrogen would be:

[1.5 atm times 0.33 = 0.5 atm]

Ideal conditions

Finally, it’s worth noting ideal conditions. Ideal conditions allow us to use pressures in thermodynamic calculations. These conditions assume:

  • Zero interactions between molecules
  • Molecules have zero size

And this is more valid for a real gas:

  • At low pressures / concentrations
  • At higher temperatures
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Introduction to Thermodynamics

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