From the combination of boyel's Charle's and Avogadro's law show that PM=dRTwhere P=pressure of gas M equals to molecular mass of gas equals to density of gas equals to temperature are equals to molar gas constant?

1 Answer
Truong-Son N.
May 15, 2018

Boyle's law was #PV = "const"#, Charles' law was #V/T = "const"#, and Avogadro's principle was #V/n = "const"#. In each of these scenarios some natural variables were held constant, such as temperature and/or pressure.

From this we find:

  • #V prop T#
  • #V prop n#

Thus, #PV prop nT#, and the constant of proportionality is #R#, the universal gas constant, so

#PV = nRT#

is the ideal gas law, where:

  • #P# is the pressure in #"atm"# if #R = "0.082057 L"cdot"atm/mol"cdot"K"#.
  • #V# is the volume in #"L"#.
  • #n# is the mols of ideal gas.
  • #T# is the temperature in #"K"#.

Now, the units of molar mass are #"g/mol"#, so if density is in #"g/L"#, then

#overbrace("g"/cancel"mol")^(M) xx (overbrace(cancel("mol"))^(n))/underbrace("L")_(V) = overbrace("g"/"L")^(d)#

and it follows that #M cdot n/V = d#, where #M# is the molar mass in #"g/mol"# and #d# is the density.

As a result, by multiplying by molar mass on both sides,

#PVM = nMRT#

#=> color(blue)(PM) = (nMRT)/V = color(blue)(dRT)#

So if a gas has a density of #"0.08988 g/L"# at STP (#0^@ "C"# and, let's say #"1 atm"#), let's solve for its molar mass.

#M = (dRT)/P#

#= ("0.08988 g/"cancel"L" cdot 0.082057 cancel"L"cdotcancel"atm""/mol"cdotcancel"K" cdot 273.15 cancel"K")/cancel"1 atm"#

#=# #"2.0145 g/mol"#

Given that this gas is a homonuclear diatomic molecule, what is it?

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