Question

Question #ccbc8

Answer 1

`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