Question

Show that the expression pRT has the same units as the pressure, and thus that the ideal gas law is dimensionally consistent.

Answer 1

Answer and Explanation:

We have given the expression $\rho RT$

Dimension of $\rho =ML^{-3}$

Dimension of R which is gas constant$=ML^2T^{-2}\Theta ^{-1}M^{-1}$

Dimension of temperature T $\Theta ^{-1}$

And dimension of pressure $ML^{-1}T^{-2}$

Now combine dimension of $\rho RT$ $=ML^{-3}ML^2T^{-2}\Theta ^{-1}M^{-1}\Theta ^{-1}=ML^{-1}T^{-2}$

So the dimension of $\rho RT$  and dimension P is same so there unit will also be same

From ideal gas equation we know that $PV=nRT$

$P=\frac{n}{V}RT=\rho RT$

As the both P and  $\rho RT$  has same dimension so they are dimensionally constant