TS EAMCET · Physics · Thermodynamics

The van der Waals' equation for a gas is \[ \left(P+\frac{a}{V^2}\right)(V-b)=n R T \] where \(P, V, R, T\) and \(n\) represent the pressure, volume, universal gas constant, absolute temperature and number of moles of a gas, respectively. \(a\) and \(b\) are constants. The ratio \(\frac{b}{a}\) will have the following dimensional formula.

A \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^2\right]\)
B \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-1} \mathrm{~T}^{-1}\right]\)
C \(\left[\mathrm{ML}^2 \mathrm{~T}^2\right]\)
D \(\left[\mathrm{MLT}^{-2}\right]\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^2\right]\)

Step-by-step Solution

Detailed explanation

van der Waals gas equation is \[ \left(p+\frac{a}{V^2}\right)(V-b)=n R T \] From principle of homogeneity Dimension of \(\frac{a}{V^2}=\) dimension of \(P\) \(\therefore\) Dimension of \(a=\left[V^2\right] \times[P]\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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