Question

A source containing a mixture of hydrogen and deuterium atoms emits red light at two wavelengths whose mean is 656.3 nm and whose separation is 0.180 nm. Find the minimum number of lines needed in a diffraction grating that can resolve these lines in the first order.

Short answer

The minimum number of lines needed are 3646 rulings.

Step-by-step solution

The resolving power

It is known that the resolving power of a grating is given by$R=Nm$ , where role="math" localid="1663051696544" $N$is the number of rulings in the grating and$m$ is the order of the lines.

The number of rulings

The resolving power is given by$R=\frac{{\lambda }_{avg}}{\Delta \lambda }$ , where${\lambda }_{avg}$ is the average of wavelengths and the separation is $\Delta \lambda$.

The resolving power also written as$R=Nm$ , where $N$is the number of rulings and$m$ is the order of the lines.

Here,${\lambda }_{avg}=656.3 \text{nm}$, $\Delta \lambda =0.18 \text{nm}$and $m=1$. So, the number of rulings can be obtained as follows:

$$\begin{gathered} Nm=\frac{{\lambda }_{avg}}{\Delta \lambda } \\ N\left(1\right)=\frac{656.3}{0.18} \\ N=3646 \end{gathered}$$

Thus, the minimum number of lines needed are 3646 rulings.