# Help on unit conversion problem

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?

the line you wrote

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

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

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.

• Thank you - this is excellent and uses what I already did to show me how to arrive to a solution for problems like these!
– Anne
Aug 28 '12 at 0:28
• Welcome! You started well but you have to write more carefully the units. Now and then write also the formulas you are going to use, even if they look very simple. This helps to cross-check the numerical calculation in each step. Longer multiplications with unit conversions come very quickly out of control (you are not the only one!) Aug 28 '12 at 0:39