What will be the ratio of wavelength of the first line to that of second line of Paschen series of H atom?

Class-12-science»Chemistry

The first line in Paschen series corresponds transition from n = 4 to n = 3, so wavelength is given as,
(RH is Rydberg's constant)
1λ1= RH 1n12-1n221λ1= RH 132-1421λ1=RH 19-1161λ1=RH 7144λ1 = 1447RH The second line in Paschen series corresponds transition from n = 5 to n = 3, so wavelength is given as,

1λ2= RH 1n12-1n221λ2= RH 132-1521λ2=RH 19-1251λ2=RH 16225λ2 = 22516RHNow, ratio of two wavelength is given as,λ1λ2 = 1447RH×16RH225 = 256175 or 1.46

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Your tool of choice here will be the Rydberg equation, which tells you the wavelength, #lamda#, of the photon emitted by an electron that makes a #n_i -> n_f# transition in a hydrogen atom.

#1/(lamda) = R * (1/n_f^2 - 1/n_i^2)#

Here

#R# is the Rydberg constant, equal to #1.097 * 10^(7)# #"m"^(-1)#

#n_i# is the initial energy level of the electron

#n_f# is the final energy level of the electron

Now, the Paschen series is characterized by #n_f = 3#.

The first transition in the Paschen series corresponds to

#n_i = 4" " -> " " n_f = 3#

In this transition, the electron drops from the fourth energy level to the third energy level.

You will have

#1/(lamda_1) = R * (1/3^2 - 1/4^2)#

The second transition in the Paschen series corresponds to

#n_i = 5 " " -> " " n_f = 3#

This time, you have

#1/(lamda_2) = R * (1/3^2 - 1/5^2)#

Now, to get the ratio of the first line to that of the second line, you need to divide the second equation by the first one.

#(1/(lamda_2))/(1/(lamda_1)) = (color(red)(cancel(color(black)(R))) * (1/3^2 - 1/4^2))/(color(red)(cancel(color(black)(R))) * (1/3^2 - 1/5^2))#

This will be equivalent to

#(lamda_1)/(lamda_2) =(1/9 - 1/25)/ (1/9 - 1/16)#

#(lamda_1)/(lamda_2) = ((25 - 9)/(25 * 9))/((16 - 9)/(16 * 9)) = 16/(25 * color(red)(cancel(color(black)(9)))) * (16 * color(red)(cancel(color(black)(9))))/7 = 16^2/(25 * 7)#

Therefore, you can say that you have

#(lamda_1)/(lamda_2) = 256/175#

What will be the ratio of the wavelength of the first line to that of the second line of Paschen Series of H atom ?A. 256: 175B. 175: 256C. 24: 27D. 15: 16

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