# 18.28 Applying the ideal gas equation to solutions

Before you read this, I suggest you read posts 16.37, 18.25 and 18.27. In post 18.27 we saw that molecules in a solution are free to move around like molecules in a gas. There are two main differences. The first is that, in a gas, movement of the molecules is constrained only by the walls of the container; in a solution, movement in an upward direction is constrained by the liquid surface (post 16.37). The second is that, in a gas, the molecules move in empty space; in a liquid, the dissolved molecules are surrounded by solvent molecules (like water).

I don’t think the first difference is very important – in both cases molecular movement is within a defined space.

Is the second difference important? Perhaps – the ideal gas equation works when collisions between molecules are very unlikely (post 18.25). In a solution, the dissolved molecules are surrounded by solvent molecules and so are very likely to collide with them. But if the dissolved molecules have a much larger mass than the solvent molecules, such collisions may have little effect on their momentum (post 17.30). A water molecule has a relative molecular mass of 18; a sugar (sucrose C12H22O11) molecule has a relative molecular mass of 342 (post 16.33). The mass of a sugar molecule divided by the mass of a water molecule is about the same as the mass of a small cannon ball divided by the mass of a golf ball. A stationary cannon ball won’t move if you throw a golf ball at it. So, I don’t think that collisions with water molecules are going to have much effect on the motion of a sugar molecule.

So let’s try applying the ideal gas equation to explaining the properties of molecules in solutions. The ideal gas equation states that pressure, p, volume V and temperature T (measured on the Kelvin scale, post 16.34) are related by

pV = nRT

where n is the number of moles of gas (post 17.48), and R is the ideal gas constant (post 18.25). For a solution, the pressure is the same as the osmotic pressure π (post 18.27). The concentration of a solution is given by

c = m/V

where V is the volume of the solution and m is the mass of stuff dissolved. In the SI system, we measure m in kg and V in m3, so c is measured in kg.m-3 (posts 16.12 and 16.13). If we substitute this information about solutions into the ideal gas equation, we get

πm/c = nRT.

Multiplying both sides of this equation by c and dividing both sides by m gives

π = cnRT/m.                    (1)

If the dissolved stuff has a relative molecular mass of M,

m = nM/1000.                    (2)

This is just saying the mass of dissolved stuff is its number of moles multiplied by the mass of a mole. But why is there a factor of 1 000 in equation 2? It’s because moles are measured in grammes (post 17.48) and mass is measured in kilogrammes (posts 16.12 and 16.13). Substituting equation 2 into equation 1 gives

π = 1000cRT/M                    (3)

Now c/M is the concentration of the solution measured in mol.m-3; so 1000c/M is the concentration measured in mol.dm-3, which is called the molarity of, CM, the solution (post 17.48). Putting this definition of CM into equation 3 gives

π = CMRT.                    (4)

Equation 4 can be used to predict the osmotic pressure of a solution. It works reasonably well for dilute solutions of stuff like sugar. But it doesn’t work for solutions that contain ions – like salt solution that contains a mixture of equal numbers of sodium (Na+) and chloride (Cl) ions (post 16.39). The ideal gas equation works when interactions (like collisions) between molecules are negligible (post 18.25). But ions with the same charge attract each other and ions with unlike charges repel each other (post 16.25) – interactions between them are not negligible! So equation 4 doesn’t work for ionic solutions. In some molecules the electric charge is not distributed evenly – like the water molecule in post 16.45. If the dissolved stuff has molecules with unevenly distributed charge, they will interact with each other in the same way as ions do. So the ideal gas 4 doesn’t work for them either.

But equation 4 does work for dilute solutions of molecules with a reasonably even charge distribution – we call these solutions ideal solutions.

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