(C) The argument of the logarithmic function must be dimensionless. Therefore,$\frac{kt}{\beta x} = 1$,which implies $\beta = \frac{kt}{x}$.Since $kt$

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(C) The argument of the logarithmic function must be dimensionless. Therefore,$\frac{kt}{\beta x} = 1$,which implies $\beta = \frac{kt}{x}$.Since $kt$ has the dimensions of energy $([ML^{2}T^{-2}])$ and $x$ is distance $([L])$,the dimensions of $\beta$ are $[\beta] = \frac{[ML^{2}T^{-2}]}{[L]} = [MLT^{-2}]$.Given that $P$ is a dimensionless quantity,the expression $P = \frac{\alpha}{\beta} \times (\text{dimensionless term})$ implies that $[P] = \frac{[\alpha]}{[\beta]}$.Since $[P] = [M^{0}L^{0}T^{0}]$,we have $[\alpha] = [\beta] = [MLT^{-2}]$.

Class 11 Physics Units, Dimensions and Measurement Dimensional Analysis, Uses and Limitations

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An expression for a dimensionless quantity $P$ is given by $P = \frac{\alpha}{\beta} \log_{e} \left( \frac{kt}{\beta x} \right)$,where $\alpha$ and $\beta$ are constants,$x$ is distance,$k$ is the Boltzmann constant,and $t$ is the temperature. Then the dimensions of $\alpha$ will be:

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A
$[M^{0} L^{-1} T^{0}]$
B
$[ML^{0} T^{-2}]$
C
$[MLT^{-2}]$
D
$[ML^{2} T^{-2}]$

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An expression for a dimensionless quantity $P$ is given by $P = \frac{\alpha}{\beta} \log_{e} \left( \frac{kt}{\beta x} \right)$,where $\alpha$ and $\beta$ are constants,$x$ is distance,$k$ is the Boltzmann constant,and $t$ is the temperature. Then the dimensions of $\alpha$ will be:

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The work done by a gas molecule in an isolated system is given by $W = \alpha \beta^{2} e^{-\frac{x^{2}}{\alpha kT}}$,where $x$ is the displacement,$k$ is the Boltzmann constant,$T$ is the temperature,and $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be:

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