Question

Calculate the dimensional formula for Rydberg's constant.

Answer 1

Step 1: Explanation for the Rydberg formula

Rydberg formula defines the relationship of frequencies in one series of lines.

Rydberg formula: $\mathrm{N}=\mathrm{R}\left[1{(\mathrm{n}+\mathrm{a})}^2-\frac{1}{{(\mathrm{m}+\mathrm{b})}^2}\right]$

Mathematically,

Rydberg constant,$\left(\mathrm{R}\right)=\frac{{\mathrm{me}}^4}{8{{\mathrm{\pi \epsilon }}_0}^2{\mathrm{h}}^2}$

$\mathrm{m}=$electron mass
$\mathrm{e}=$electron charge
${\mathrm{\epsilon }}_0=$the permittivity of free space
$\mathrm{h}=$Planck’s Constant

Step 2: Determine the dimensional formula of the Rydberg constant

So, if we enter all of the dimensional formulas into the preceding equation, we get

Dimensional formula of Rydberg constant is,

$$\begin{gathered} =\left[\frac{\left[{\mathrm{M}}^1{\mathrm{L}}^0{\mathrm{T}}^0\right]{\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{A}}^1{\mathrm{T}}^1\right]}^4}{\left[{\left({\mathrm{M}}^{-1}{\mathrm{L}}^{-3}{\mathrm{T}}^4{\mathrm{A}}^2\right)}^2\right]\left[{\left({\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-1}\right)}^2\right]}\right] \\ ={\mathrm{M}}^0{\mathrm{L}}^{-1}{\mathrm{T}}^0 \end{gathered}$$ .

The SI unit of the Rydberg constant is ${\mathrm{m}}^{-1}$.

Hence, the dimensional formula for Rydberg's constant is ${\mathrm{M}}^0{\mathrm{L}}^{-1}{\mathrm{T}}^0$.