Find out the unit and dimensions of the constants \(a\) and \(b\) in the van der Waal's equation…

Find out the unit and dimensions of the constants \(a\) and \(b\) in the van der Waal's equation \(\left(p+\frac{a}{V^{2}}\right)(V-b)=R T\), where \(p\) is pressure, \(v\) is volume, \(R\) is gas constant, and \(T\) is temperature.
  1. $A: \mathrm{Nm}^{4}, B: m^{3}$
  2. $A: m^{3}, B: N m^{4}$
  3. $A: N^{2} m^{3}, B: m^{4}$
  4. $A: \mathrm{N m}^{3}, B: m^{3}$

Solution

We can add and subtract only like quantities.
\(\Rightarrow\) Dimensions of \(P=\) Dimensions of \(\frac{a}{V^{2}}\) ...(i)
and dimensions of \(v=\) Dimensions of \(b\) ...(ii)
From (i),
Dimensions of \(a=\) Dimensions of \(P \times\) Dimensions of \(V^{2}\) \([a]=\left[M^{1} L^{-1} T^{-2}\right] \times\left[L^{3}\right]^{2}=\left[M^{1} L^{5} T^{-2}\right]\)
Unit of \(a=\) Unit of \(p \times\) Unit of \(V^{2}=\frac{N}{\mathrm{~m}^{2}} \times \mathrm{m}^{6}=\mathrm{N} \mathrm{m}^{4}\)
From (ii), \([b]=[V]=\left[M^{\circ} L^{3} T^{0}\right]\)
So unit of \(b=\) Unit of \(V=\mathrm{m}^{3}\) *

Asked in: JEE Mains - Units and Dimensions - Test 1

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