Question #ccbc8

1 Answer
Jul 31, 2016

Answer:

#R = 0.082("atm" * "dm"^3)/("mol" * "K")#

Explanation:

Your starting point here will be the ideal gas law equation

#color(blue)(|bar(ul(color(white)(a/a)PV = nRTcolor(white)(a/a)|)))" "#

Here you have

#P# - the pressure of the gas
#V# - the volume it occupies
#n# - the number of moles of gas
#R# - the universal gas constant
#T# - the absolute temperature of the gas

Rearrange the above equation to solve for #R#

#PV = nRT implies R = (PV)/(nT)#

Now, notice that the problem doesn't provide you with the number of moles of nitrogen gas present in your sample. This means that you're going to have to improvise a bit.

As you know, the number of moles present in a sample of a given compound is equal to the mass of the sample divided by the molar mass of the compound.

If you take #m# to be the mass and #M_M# the molar mass of nitrogen gas, you can say that

#n = m/M_M#

Plug this into the above equation to get

#R = (PV)/(m/M_M * T) = (PV)/(m * T) * M_M#

Now, the problem tells you that the gas has a density of #"1.250 g dm"^(-3)#. As you know, density is defined by mass divided by volume.

You already used #m# as the mass of the sample, so you can say that you have

#rho = m/V implies V/m = 1/(rho)#

Plug this into the equation to get

#R = P/T * 1/(rho) * M_M#

Nitrogen gas, #"N"_2#, has a molar mass of #"28.0134 g mol"^(-1)#. Plug in the values given to you to calculate the value of #R# -- do not forget to convert the temperature from degrees Celsius to Kelvin

#R = "1 atm"/((273.15 + 0)"K") * 1/(1.250 color(red)(cancel(color(black)("g"))) "dm"^(-3)) * 28.0134 color(red)(cancel(color(black)("g"))) "mol"^(-1)#

#color(green)(|bar(ul(color(white)(a/a)color(black)(R = 0.082 ("atm" * "dm"^3)/("mol" * "K"))color(white)(a/a)|)))#

I'll leave the answer rounded to two sig figs, but don't forget that you only have one sig fig for the temperature and pressure of the sample.

The actual value of #R# listed using these units is

#R ~~ 0.0820575("atm" * "dm"^3)/("mol" * "K")#

so this is an excellent result.

http://www.cpp.edu/~lllee/gasconstant.pdf