Q:

How can I determine density of a gas only given temperature?

I have a homework problem where exhaust is traveling through an exhaust system (assumed to be air for simplicity) from and engine and then released into the atmosphere. The exhaust is at a temperature of 250F. I need to find the pressure at the beginning of the exhaust system.
I've been able to set this up and created an equation to solve for the pressure. I assumed that the pressure of the atmosphere where it it exits is 0 atm. But the problem is that the pressure depends on the density of the exhaust (air). I was only given temperature of the exhaust in the problem and I need a second property to determine density. Is there another way I can determine density? I am also given the volumetric flow rate and the area of the piping system

A:

The key's is the Bernoulli's equation for the compressible flow:
$$\frac{v^2}{2} + \frac{p}{\rho} + u = \text{const}$$
$u$ is internal energy per unit mass, or using enthalpy $h$ per unit mass:
$$\frac{v^2}{2} + h = \text{const}$$
The other equation to find $v$ you'll get from the definition of the volumetric flow.
You have two equations to solve the system. I let it to you to find the $h$ for the ideal gas, you'll find that it depends only on the temperature.
So you'll find the temperature in the engine. To find the pressure in the engine you'll need also the density. Here mass conservation comes in hand:
$$\rho v S = \rho Q = const$$
where $Q = vS$ is the volumetric flow rate. All you need is the density at the exhaust which you find form the ideal gas law, knowing the exhaust temperature and pressure (1 atm or 0 atm gauge).

