Units & Dimensions

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75 of 149

Question 75 of 149

(P+aV2)(V−b)=RT\left(P+\dfrac{a}{{V}^{2}}\right)\left(V-b\right)=RT represents the equation of state of some gases. Where PP is the pressure, VV is the volume, TT is the temperature and a,b,Ra,b,R are the constants. The physical quantity, which has dimensional formula as that of b2a\dfrac{{b}^{2}}{a} , will be :

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Answer: (C) Compressibility

According to the principle of dimensional homogeneity, the dimensions of each terms of a dimensional equation on both sides are the same.
So, dimension of bb and VV will be same, [b]=[V]\left[b\right]=\left[V\right]
And dimension of aV2\dfrac{a}{{V}^{2}} and PP will be same, [aV2]=[P]\left[\dfrac{a}{{V}^{2}}\right]=\left[P\right]
From above two relations, we have [ab2]=[P]\left[\dfrac{a}{{b}^{2}}\right]=\left[P\right]
∴[b2a]=1[P]=[ML−1T−2]−1∴\left[\dfrac{{b}^{2}}{a}\right]=\dfrac{1}{\left[P\right]}={\left[M{L}^{-1}{T}^{-2}\right]}^{-1}
Thus, [M−1L1T2]\left[{M}^{-1}{L}^{1}{T}^{2}\right] is the dimension of compressibility.

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