Help on unit conversion problem

Question

This is a problem from school. I will show my attempt.

The question:

"The gas constant for dry air R is 287 $\frac{m^2}{s^2*K}$. Assuming the temperature is 330 K and the pressure is 1050 hPa, what is the atmospheric density."

The professor said DO NOT produce an answer by finding a formula, but to use the magic of unit conversion to try to solve things.

I know density is measured in kg/m^3 or thereabouts so I tried the following:

1050 hPA = 105, 000 Pa

1 Pa = 1 kg/m*s^2

105,000 $\frac{kg}{m*s^2}$ * 330 K * 287 $\frac{m^2}{s^2*K}$.

This cancels some units... but not enough...in fact it cancels just K, so far as I understand, far from what I need for my density unit.

Any ideas on what Im doing foolishly here?

Answer 1

the line you wrote

105,000 $\frac{kg}{m*s^2}$ * 330 K * 287 $\frac{m^2}{s^2*K}$

has to read in fact

$\frac{105,000 \frac{kg}{m*s^2} }{ 330 K * 287 \frac{m^2}{s^2*K}} = 1.11 \frac{kg}{m^3}$.

This comes from the gas law

$p=\rho \ R \ T $

where $p$ is the air pressure and $\rho$ is the air density. Solving for $\rho$ you get

$\rho =\frac{p}{R T} $

from which the numerical solution follows.