Question

The speed v in a gas can reasonably be dependent on the pressure p, the density p, and the volume V of the gas. Using dimensional analysis, determine the exponents l,m and n in the formula v=Cp^lp^mV^n

Answer 1

We have:

v=\rho^lp^mV^n,\\\space\\ [\text{m/s}]=[\text{kg/m}^3]^l·[\text{kg/(s}·\text m)]^m·[\text{m}^3]^n,\\ [\text{m}]^1[\text s]^{-1}=[\text{kg}]^{l+m}[\text{m}]^{-3l-m+3n}[\text s]^{-m}.

Comparing the powers, we see that

$$l+m=0,\\ -3l-m+3n=1,\\ -m=-1.\\\space\\ l=-1,\\ m=1,\\ n=-1/3.\\\space\\ v=\dfrac{p}{\rho\sqrt[3]{V}}.$$