Question 135: Units , Dimensions and Error Analysis

Question

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

Accepted Answer

Given, $P=\frac{\alpha}{\beta} \log _{e}\left[\frac{k t}{\beta x}ight]$
The logarithmic term is dimensionless.
Thus, $[k t / \beta x]$ is also dimensionless.
i.e. $\frac{[k][t]}{[\beta][x]}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}ight]$ .......(i)
We have, $E=k t$
Thus, Eq. (i) becomes,
$$
\begin{array}{r}
\frac{\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}ight]}{[\mathrm{\beta}]\left[\mathrm{L}^{1}ight]}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}ight] \\
{[\beta]=\left[\mathrm{MLT}^{-2}ight]}
\end{array}
$$
Since, $P$ is also a dimensionless quantity.
$$
\begin{aligned}
&\frac{[\alpha]}{[\beta]}=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}ight] \\
&{[\alpha]=[\beta]\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}ight]} \\
&{[\alpha]=\left[\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}ight]=\left[\mathrm{MLT}^{-2}ight]}
\end{aligned}
$$

Answer

[M0 L$$-$$1 T0]

Answer

[M L0 T$$-$$2]

Answer

[M L T$$-$$2]

Answer

[M L2 T$$-$$2]

Units , Dimensions and Error Analysis

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

Choose the correct answer: