Now having all these individual gas laws can get confusing, even though they are fairly simple. But what happens when all three variables are changing? When T, P and V change?

We need to combine all of these laws together.

Since:

V = a/P, V = b T, and V = c n, we can combine all of the constants into one constant and combine all of these equations:

V µ T n / P or VP/Tn = constant so that

V₁ P₁ / T₁ n₁ = V₂ P₂ / T₂ n₂

Those values that don’t change from one condition to the next will simply cancel out and we'll get the appropriate relationships between the variables.

For example: We blow up our 2.56 L balloon at sea level on a summer day (P = 1 atm, T = 32.0 ºC). Then we take it up to the top of Mt Everest where P = 0.300 atm, and T = -27.0 °C. What is the new volume?

moles remain constant so n cancels on both sides and we are left with:

V₁ P₁ / T₁ = V₂ P₂ / T₂

Solve for V₂ = V₁ P₁ T₂ / T₁ P₂ =

So if VP/Tn = constant, we can name this constant R (the ideal gas constant) and measure it’s value = 0.082058 L atm/ mol K (note the units. This means that P must be in atm, V in L and T in K)

Thus we can write: VP / Tn = R or

PV = nRT which is the ideal gas law.

Thus if we know any three of the four variables, we can calculate the other.

We can also use this equation to determine the molecular mass of a gas:

remember that molecular mass is gram/mole, so if we put a fixed mass of gas into a manometer of known volume and temperature, we can determine the number of moles:

n = PV/RT

Dividing the mass by the moles gives us molar mass:

m/n = mRT/PV = M

example: ex 5.12 a

If 0.440 g of a gas occupies 179 ml at a pressure of 741 mmHg at a T of 86.0ºC, what is the molar mass of this gas?

Gas density:

density is mass/volume, thus if we take our new equation and solve for m/V we get:

m/V = MP / RT = d

Example: exercise 5.14a

What is the density of ethane gas (C₂H₆) in g / L at 15.0 ºC and 748 Torr?