Units & Dimensions

JEE Main level, single correct, medium

94 of 149

Question 94 of 149

An expression for a dimensionless quantity PP is given by P=αβlog⁡e(kTβx)P=\dfrac{\alpha }{\beta }{\log }_{e}\left(\dfrac{kT}{\beta x}\right); where α\alpha and β\beta are constants, xx is distance; kk is Boltzmann constant and TT is the temperature. Then the dimensions of α\alpha will be

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Answer: (C) [MLT−2]\left[{MLT}^{-2}\right]

Given p=αβln⁡(kTβx)p=\dfrac{\alpha }{\beta }\ln \left(\dfrac{kT}{\beta x}\right)
We know that expression inside log is dimensionless.
Therefore,[kTβx]\left[\dfrac{kT}{\beta x}\right] is dimensionless
⇒[β]=[kTx]=ML2T−2K−1×K×L−1=MLT−2\Rightarrow \left[\beta \right]=\left[\dfrac{kT}{x}\right]={ML}^{2}{T}^{-2}{K}^{-1}\times K\times {L}^{-1}={MLT}^{-2}
Since pp is dimensionless ∴[p]=[αβ]=M0L0T0∴\left[p\right]=\left[\dfrac{\alpha }{\beta }\right]={M}^{0}{L}^{0}{T}^{0} ⇒[α]=[β]\Rightarrow \left[\alpha \right]=\left[\beta \right] ⇒[α]=MLT−2\Rightarrow \left[\alpha \right]={MLT}^{-2}

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