okay let's check out a problem regarding the molar mass of a gas the
question asks an evacuated glass vessel weighs 50 point zero grams when empty
148 point 0 grams when completely filled with a liquid of density 0.98 grams per
milliliter and 50 point five grams when filled with an ideal gas at 760
millimeters mercury at 300 Kelvin what is the molar mass of the gas so
certainly we're going to be dealing with ideal gases and all the equations that
relate to them so if you want to review check out some of my tutorials on this
subject and when you're ready give it a try
okay first let's do some very basic
calculations to get the values that are going to be relevant first we're told
that we have some liquid and we know that the vessel filled with liquid is
148 grams and then without the liquid is 50 grams and so simply by subtraction we
know that the liquid has a mass of 98 grams that is the mass of the liquid
that completely fills this glass vessel then we know that the density of the
liquid is 0.98 grams per milliliter so if we want to get the volume of the
liquid we know that 98 grams divided by 0.98 grams per milliliter is going to
give us a hundred milliliters and so that's important because if we have a
hundred milliliters of liquid and that liquid completely fills the vessel that
means that the volume of the vessel itself is a hundred milliliters which is
relevant information because now we're going to look at the gas we're told that
when the vessel is filled with this gas it has a mass of 50 point five grams and
then again when empty 50 grams so the mass of the gas is 0.5
grams now we have all the information we need to start using the ideal gas law we
know that the vessel has a volume of 100 milliliters
we know that the internal pressure is 760 millimeters mercury
we know that the temperature is 300 Kelvin and then we do have a mass here
of 0.5 grams and we're going to use this mass in conjunction with the moles value
in the ideal gas law so let's get our ideal gas law as we know that is PV
equals NRT now the thing is we're going to want to do something with moles so
that we can express this in terms of mass because we have a mass and we want
to figure out molar mass so we know that molar mass is equal to mass over moles
right if we have some grams value and we have some moles value number of grams
over number of moles that's grams per mole that's the molar mass so let's
rearrange that expression and plug it in we're moles is so if if Big M equals
little m over N then we can solve for moles and we can say that moles is equal
to little m over Big M so moles is equal to mass over molar mass and we can then
plug that in instead of moles in our ideal gas law then since we are looking
for molar mass let's solve for molar mass let's solve for big m in this
equation all we have to do is multiply both sides by molar mass and then divide
both sides by pressure and volume so Big M equals little m RT over P V then we
just plug in what we know so for mass we know we have 0.5 grams of the gas then
we have the gas constant which as always is 0.0821 liters atmospheres per mole
Kelvin in this kind of context and then for T we have 300 Kelvin that's given to
us and then for pressure let's be sure to convert this into atmospheres because
we have atmospheres in our gas constant so luckily 760 we should remember is
equal to one atmosphere for millimeters of mercury
and the volume is what we solved for earlier we know that we have a 100
milliliter vessel again here let's change to liters to match our gas
constant so that's 0.1 liters so we plug that all
into our calculator and we get 123 grams per mole as the molar mass of this gas