Question

The path length of a cell for infrared spectroscopy can be measured by counting interference fringes (ripples in the transmission spectrum). The following spectrum shows 30interference maxima between 1906and$\mathbf{698}\mathit{c}{\mathit{m}}^{\mathbf{-}\mathbf{1}}$obtained by placing an empty KBrcell in a spectrophotometer.

The fringes arise because light reflected from the cell compartment interferes constructively or destructively with the unreflected beam.

If the reflected beam travels an extra distance, it will interfere constructively with the unreflected beam. If the reflection path length is $\mathit{\lambda }\mathbf{/}\mathbf{2}$, destructive interference occurs. Peaks therefore arise when $\mathit{m}\mathit{\lambda }\mathbf{/}\mathbf{2}\mathbf{=}\mathbf{2}\mathit{b}$and troughs occur when, where $\mathit{m}\mathit{\lambda }\mathbf{/}\mathbf{2}\mathbf{=}\mathbf{2}\mathit{b}$is an integer. If the medium between KBr theplates has refractive index n, the wavelength in the medium is l/n, so the equations become $\mathit{m}\mathit{\lambda }\mathbf{/}\mathit{n}\mathbf{=}\mathbf{2}\mathit{b}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{m}\mathit{\lambda }\mathbf{/}\mathbf{2}\mathit{n}\mathbf{=}\mathbf{2}\mathit{b}$. The cell path length can be shown to be given by

$\mathit{b}\mathbf{=}\frac{\mathbf{N}}{\mathbf{2}\mathbf{n}}\mathbf{\times }\frac{{\mathbf{\lambda }}_{\mathbf{1}}{\mathbf{\lambda }}_{\mathbf{2}}}{{\mathbf{\lambda }}_{\mathbf{2}}\mathbf{-}{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{=}\frac{\mathbf{N}}{\mathbf{2}\mathbf{n}}\mathbf{\times }\frac{\mathbf{1}}{{{\displaystyle \stackrel{\mathbf{~}}{\mathbf{v}}}}_{\mathbf{1}}\mathbf{-}{\stackrel{\mathbf{~}}{\mathbf{v}}}_{\mathbf{2}}}$

where Nmaxima occur between wavelengths${\mathit{\lambda }}_{\mathbf{1}}$and${\mathit{\lambda }}_{\mathbf{2}}$. Calculate the path length of the cell that gave the interference fringes shown earlier.

Short answer

The cell path length b = 0.0124cm

Step-by-step solution

Step 1:

The path length is a scalar quantity that is the distance travelled by an object. A path length can never be zero or negative for a moving object.

Calculating the pathway

The cell pathlenght can be calculated by formula:

$$\begin{gathered} b=\frac{N}{2n}.\frac{1}{{{\displaystyle \tilde{v}}}_1-{\tilde{v}}_2} \\ b=\frac{30}{2.1}.\frac{1}{1906c{m}^{-1}-698c{m}^{-1}} \\ b=0.0124cm \end{gathered}$$