Question

An atom has x energy level, then total number of lines in its spectrum are :

Answer 1

Number of spectrum lines of an atom

1. A spectral line is a dark or bright line in a spectrum. this is because of the emission or absorption of light in that range where the lines are formed.
2. We know that there n number of levels exists between the nth quantum level to the ground state level (n=1).
3. According to Bohr's quantum theory, when an atom has n energy level then the number of lines in its spectrum is defined by the form, ${}^{n}C_2=\frac{n\left(n-1\right)}{2}.$

Finding the number of lines

We know the number of spectrum lines in n energy level is ${}^{n}C_2=\frac{n\left(n-1\right)}{2}.$

So, for x energy level, the number of spectral lines is ${}^{x}C_2=\frac{x\left(x-1\right)}{2}.$

or, ${}^{x}C_2=\frac{x\left(x-1\right)}{2}=1+3+5+7+.....+\left(x-1\right)$

Now, $$\begin{gathered} At,n=0,\frac{n\left(n-1\right)}{2}=0. \\ At,n=1,\frac{n\left(n-1\right)}{2}=0. \\ At,n=2,\frac{n\left(n-1\right)}{2}=1. \\ At,n=3,\frac{n\left(n-1\right)}{2}=3. \\ At,n=4,\frac{n\left(n-1\right)}{2}=5. \\ .. \\ So,{}^{x}C_2=\frac{n\left(n-1\right)}{2}=1+3+5+7+.....+\left(x-1\right). \end{gathered}$$

Therefore, the total number of lines in its spectrum is $1+3+5+7+.....+\left(x-1\right).$