Compressibility Factor and Generalized Compressibility Chart From my understanding, the compressibility factor is defined by
$$Z=\frac{pv}{RT}=\frac{p\bar{v}}{\bar{R}T}$$
It can also be defined by the ratio of the real molar volume of a gas to the ideal molar volume of a gas at the same temperature and pressure. Essentially it corrects for the deviation of a real gas from an ideal gas.
On a generalized compressibility chart, the compressibility $Z$ is plotted as a function $f=f(p_R,T_R)$ of the reduced pressure and temperature. I don't understand why exactly; it would be nice if someone could explain that a little more.
Another thing I'm confused about is the psuedoreduced specific volume, given by
$$v'_{R}=\frac{\bar{v}}{\bar{R}T_c/p_c}$$
Why don't we use the reduced specific volume as opposed to the psuedoreduced specific volume?
In my thermodynamics class, I was presented with the following question:
Determine the temperature, in °C, of air at 30 bar and a specific volume of 0.013 $m^3/kg$. Use compressibility chart.
I was confused about how to figure out this problem since I can't find the reduced temperature without the actual temperature of the air.
You start out with an initial guess for the temperature.  I suggest starting out using the value that you would calculate using the ideal gas law.  You then,

Calculate the reduced temperature and pressure
Get z at this reduced temperature and pressure
Use these to calculate a value for the specific volume.
If the value for the calculated specific volume does not match 0.013, adjust your guess for the temperature and go back to step 1.

Continue doing this until the calculated specific volume matches the prescribed value of 0.013