For He+ ion, the wave number (`barv`)associated with the Balmer transition, n = 4 to n = 2 is given by:
`bar "v" = 1/lambda = "RZ"^2(1/"n"_1^2 - 1/"n"_2^2)`
Where
n1 = 2
n2 = 4
Z = atomic number of helium
`bar "v" = 1/lambda = "R"(2)^2(1/4 - 1/16)`
`= 4"R" ((4-1)/16)`
`=bar "v" = 1/lambda = (3"R")/4`
`=> lambda = 4/(3"R")`
According to the question, the desired transition for hydrogen will have the same wavelength as that of He+.
`=> "R"(1)^2[1/"n"_1^2 -1/"n"_2^2] = (3"R")/4`
`[1/"n"_1^2 - 1/"n"_2^2] = 3/4 .....(1)`
By hit and trail method, the equality given by equation (1) is true only when
n1 = 1 and n2 = 2.
∴ The transition for n2 = 2 to n = 1 in hydrogen spectrum would have the same wavelength as Balmer transition n = 4 to n = 2 of He+ spectrum.
Text Solution
`2 rarr 1``5 rarr 3``4 rarr 2``6 rarr 4`
Answer : C
Solution : `bar(v)_(H) = R[(1)/(1^(2))-(1)/(2^(2))] cm^(-1)` ...(i) <br> `bar(v)_(He^(+)) = R(2)^(2)[(1)/(n_(1)^(2))-(1)/(n_(2)^(2))]` ....(ii) <br> Now, `bar(v)_(He^(+)) = R[(1)/(1^(2))-(1)/(2^(2))] = R xx ((2)^(2))/((2)^(2)) [(1)/(1^(2))-(1)/(2^(2))]` <br> `= R(2)^(2)[(1)/(2^(2))-(1)/(4^(2))]` ...(iii) <br> Comparing (ii) and (iii), it is clear that <br> `n_(2) = 4, n_(1) = 2`.