You will work with the ideal gas law and your understanding of density to apply these tools to working with molar mass problems.
Now we will examine equation 1 above a bit further. We will let the proportionality constant be R rather than Ca and then we have P = R(nT/V) or PV = nRT. This relationship, PV = nRT, is called the combined gas law or the ideal gas law. The equation PV = nRT is also called the ideal gas (IG) equation of state (EOS). An equation of state is an equation relating state variables, and the IG EOS is the simplest EOS. R is called the ideal gas constant. The value of R will depend upon the units of the variables. At STP (standard temperature and pressure) conditions, one mole of a gas will occupy 22.414 liters (called the standard molar volume). So solving the ideal gas EOS for R gives R = PV/nT and substituting in the standard conditions gives R = 0.082057 Latm/(molK). We can use this equation just as we used equation 1 to solve problems involving changes in state variables.
The density of a gas is given by D = mass/volume so we can rearrange our IG EOS to obtain an equation involving density. We know that n = m/M (mass/molecular weight) so we have PV = (m/M)RT or PV = mRT/M. Now if we divide both sides by V we get: P = (m/V)RT/M and we recognize that m/V is the density, D, so we have
P = DRT/M or D = MP/RT ...................eq 5
This EOS with density has the obvious example to how hot-air balloons work, as your text explains. Many people don't know that such balloons are totally open at the bottom of the balloon so that gas can escape from the balloon as it is heated. When the heater is turned on, the gas needs to expand (V is directly proportional to T), but instead it escapes so the density of the gas decreases (V is the same, but m is less). We know that the less-dense objects rise and objects with higher densities fall so the balloon functions as needed in obedience to this EOS.
We can easily see another application of equation 5 by solving it for the molar mass:
M = DRT/P .................................eq 6
If we experimentally measure the density of an unknown gas at a particular T and P, then we can calculate the molecular weight of the unknown gas.
Always start these problems with the IG EOS and derive the needed equations
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