Dimensional Analysis Question [closed]

Question

First of, I would like to say that I have tried this question, and have my answer as well, just not sure such a method of obtaining the answer is valid or not, therefore trying to look for help here. By the way, I am new here, prior apology for any violation of rules and regulation (and poor question expression as well).

Here's the question:

In the gas equation:

$(p + \frac{a}{V^2}) (V-b) = RT$,

where $p$, $V$, and $T$ are the pressure, volume and gas temperature, respectively.

Determine the dimensions of $a$ and $b$.

Note: The $R$ is not specified, suppose it should be a constant.

Here's my approach of answer:

Due to possibility of temperature being at absolute 0, therefore $RT$=0 is possible.

With that, I let $(p + \frac{a}{V^2})=0$, $(V-b)=0$

(Dimensional analysis working)

The answer turns out to be:

[$a$]=(M) (L$^5$) (T$^{-2}$), [$b$]=L$^3$

Just wondering such approach of answer is valid or not, if not, what should it be. Replies are greatly appreciated. By the way, how to use MathJax here? I have very very limited of knowledge on html as well, thank you.

Answer 1

Approach sort of valid.

See the dimensions of b would be those of V, but for another reason -- you needn't have put zero at RHS. The reason is that only quantities of the same dimension can be added or subtracted to each other. It doesn't matter if RT can equal zero or not. Similarly dimensions of al(V^2) would be those of p.

Check out Rules of Evaluating Equations here: http://go.hrw.com/resources/go_sc/ssp/HK1BSW23.PDF

Answer 2

I think R is intended to be the well-known ideal gas constant, which has dimensions of energy/time (joules per mole per kelvin in SI units). So in this equation the left-hand and right-hand sides both have dimensions of energy.

However, you don't need to know that to answer the question. As Gautam Khare stated, you can only add terms with the same dimensionality, so b must have the same dimensions as V, and a/V^2 must have the dimensions of P.