Chemistry A-Level question on molecular mass

Calculate the relative molecular mass of a gas which has a density of 2.615gdm^-3 at 298K and 101 kPa

Explanation please?

This requires some equation manipulation

PV=nRT and n = m / MM and density = mass / volume (d = m / V

P = pressure, V = volume, n = number of moles, R = universal gas constant, T = temperature, MM = relative molecular mass, d = density, m = mass.

Right ok so:

d = m/V

V = m/d

PV=nRT

Pm/d = nRT

m/n = RTd/P

Now remember n = m / MM

So MM = m/n

Therefore:

MM = RTd/P

MM = 8.314 J K-1 mol-1 x 298K x 2.615 g dm^-3 / 101000 pa

We have a unit problem here.

1 J= 1kg m^2 s-1

1 pa = 1 kg m-1 s^-2

So we need to convert our density into m^-3

1dm^3 = 0.001 m^3 Therefore 1 dm^-3 = 1000 m^3

density = 2615 g m^-3

MM = 8.314 J K-1 mol-1 x 298K x 2615 g m^-3 / 101000 pa = 64.1 g mol-1

Therefore the relative molecular mass of the gas is 64.1 g/mol


======== follow up ====

KennyB is wrong. MW (or MM) = RTd/P NOT MW = RT/dP

You need three equations:

1) the gas law: n = PV/RT
and
2) the relationship between moles and mass: n = (m/MW)
and
3) the density equation: d = m/V

Rearrange: MW = RT/dP

(note that the volume "vanishes" from the equation.

Now
T = 298 K
P = 101 kPa
d = 2.615g/dm^3

You need to use an R in units of kPa-dm^3/deg-mole and your answer will come out in g/mole.