Question

The equation for real gas is given by $ \left( P + \frac{a}{V^2} \right)(V - b) = RT $, where $ P $, $ V $, $ T $, and $ R $ are the pressure, volume, temperature and gas constant, respectively. The dimension of $ ab $ is equivalent to that of:

• A. Planck's constant
• B. Compressibility
• C. Strain
• D. Energy density

Hint:

To solve for dimensional analysis problems, break down each term into its basic dimensions and multiply accordingly.

Answer 1

The Correct Option is B

From the given equation \( \left( P + \frac{a}{V^2} \right)(V - b) = RT \), we have the following dimensions for each variable: \[ [a] = \left[ P \right] \left[ V \right]^2 = ML^{-1}T^{-2}L^2 = M L T^{-2} \] \[ [b] = [V] = L^3 \] Now, \( [ab] = (M L T^{-2})(L^3) = M L^4 T^{-2} \). 
Thus, the dimensions of \( ab \) correspond to the dimension of compressibility.