Units & Dimensions

JEE Main level, single correct, medium

69 of 149

Question 69 of 149

In the equation [X+aY2][Y−b]=RT,X\left[X+\dfrac{a}{{Y}^{2}}\right][Y-b]=RT,X is pressure, YY is volume, RR is universal gas constant and TT is temperature. The physical quantity equivalent to the ratio ab\dfrac{a}{b} is:

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Answer: (B) Energy

X&\dfrac{a}{{Y}^{2}} have the same dimensions. Y&b also have the same dimensions.
Therefore,
aY2=ML−1T−2⇒a=ML−1T−2×L6=ML5T−2\dfrac{a}{{Y}^{2}}=M{L}^{-1}{T}^{-2} \Rightarrow a=M{L}^{-1}{T}^{-2}\times {L}^{6}=M{L}^{5}{T}^{-2}
and b=L3b={L}^{3}.
So, the dimensions of ab\dfrac{a}{b} is,
ab=ML5T−2L3=ML2T−2\dfrac{a}{b}=\dfrac{M{L}^{5}{T}^{-2}}{{L}^{3}}=M{L}^{2}{T}^{-2}.
This is the dimensional form of energy. Hence, this is the right option.

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