Question

Ratio of the wavenumber of the first line of Lyman
series of hydrogen atom to the first line of Balmer
series of He+ ion will be

A. 5:2
B. 9:5
C. 27:20
D. 7:5​

Answer 1

To Find :

• The Ratio of wavenumber of the first line of Lyman series of hydrogen atom to the first line of Balmer series of He⁺ ion

Solution :

Wavenumber of a spectral line is given by ,

$\\ \star \: {\boxed{\purple{\sf{ \frac{1}{ \lambda} =RZ^2 \bigg( \dfrac{1}{ {n_1}^{2} } - \dfrac{1}{ {n_2}^{2} } \bigg) }}}}$

Here ,

• R is Rydberg's constant
• Z is atomic number
• n₁ and n₂ are energy levels (n₂ > n₁)

First let us calculate the wavenumber of first line of Lyman series of hydrogen atom.

• For first line of lyman series , n₁ = 1 and n₂ = 2
• Z = 1 {Atomic number of hydrogen atom}
• R {Constant}

Substituting the values ;

$\\ : \implies \sf \: \dfrac{1}{ \lambda_1} = R \times {1}^{2} \bigg( \dfrac{1}{ {1}^{2} } - \dfrac{1}{ {2}^{2} } \bigg)$

$\\ : \implies \sf \: \dfrac{1}{ \lambda_1} = R \times 1 \bigg( \dfrac{1}{1} - \dfrac{1}{4} \bigg)$

$\\ : \implies \sf \: \dfrac{1}{ \lambda_1} = R \bigg( \dfrac{3}{4} \bigg)$

$\\ :\implies{\underline{\boxed{\red{\sf{\overline{\upsilon}_1= \dfrac{3R}{4} }}}}} \: \bigstar$

[ Since wavenumber is equal to the reciprocal of wavelength]

Now , Calculating the wavenumber of first line of Balmer series of He⁺ ion.

• For first line of balmer series Balmer series , n₁ = 2 and n₂ = 3.
• Atomic number of He⁺ ion = 2

Substituting the values ;

$\\ : \implies \sf \: \dfrac{1}{ \lambda_2} = R \times 2^2 \bigg( \dfrac{1}{ {2}^{2} } - \dfrac{1}{ {3}^{2} } \bigg)$

$\\ : \implies \sf \: \dfrac{1}{ \lambda_2} = R \times 4 \bigg( \dfrac{1}{4} - \dfrac{1}{9} \bigg)$

$\\ : \implies \sf \: \dfrac{1}{ \lambda_2} = 4R \bigg( \dfrac{5}{36} \bigg)$

$\\ :\implies{\underline{\boxed{\red{\sf{\overline{\upsilon}_{2} = \dfrac{20R}{36} }}}}} \: \bigstar$

Now , Calculating the ratio ;

$\\ : \implies \sf \: \dfrac{ \overline{ \upsilon}_1}{ \overline{ \upsilon}_2} = \dfrac{ \frac{3R}{4} }{ \frac{20R}{36} }$

$\\ : \implies \sf \: \dfrac{ \overline{ \upsilon}_1}{ \overline{ \upsilon}_2} = \dfrac{ \frac{3}{4} }{ \frac{20}{36} }$

$\\ : \implies \sf \: \dfrac{ \overline{ \upsilon}_1}{ \overline{ \upsilon}_2} = \dfrac{ 3 \times 36}{ 20 \times 4 }$

$\\ : \implies{\underline{\boxed{\pink{\sf{ \: \dfrac{ \overline{ \upsilon}_1}{ \overline{ \upsilon}_2} = \dfrac{ 27}{ 20 }}}}}} \: \bigstar$

Hence ,

• The required ratio is 27 : 20. So , Option(3) is the required answer.