I am a chemical engineering student learning about dimensionless quantities. This is a practice question that I am trying.
The Van der Waals equation of state can be used to predict the behaviour of non-ideal gas, and it reads $$ \left( P + \frac{n^2a}{V^2}\right) (V - nb)= nRT, $$ where
- $P$ is the pressure, in $\rm N/m^2$,
- $n$ is number of moles, in $gmol$,
- $V$ is the volume, in $\rm m^3$,
- $T$ is the temperature, in $\rm K$,
- $R$ is the molar gas constant, in $atm.L / gmol.K$, and
- $a$ and $b$ are constants.
I am trying to show that this equation is dimensionally consistent. But first I must find the units of $a$ and $b$.
I learned that units on the left must be equals to units on the right if it is dimensionally consistent. Knowing that, I found that $nRT= atm.L$
However, now I get stuck because on the left hand side of the equation, it’s something multiply by something. Meaning the whole thing has a unit of $atm.L$ . But how do I find them individually?
In short, It’s wrong to say that $( P + \frac{n^2a}{V^2})$ has a unit of $atm.L$ so how do I find units of a and b ? How do I know what is the unit of $( P + \frac{n^2a}{V^2})$ and also the other part of the equation?