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प्रश्न
A group of hydrogen atoms are prepared in n = 4 states. List the wavelength that are emitted as the atoms make transitions and return to n = 2 states.
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उत्तर
There will be three wavelengths.
(i) For the transition from (n = 4) to (n = 3) state
(ii) For the transition from (n = 3) to (n = 2) state
(iii) For the transition from (n = 4) to (n = 2) state
Let `(lamda_1)` be the wavelength when the atom makes transition from (n = 4) state to (n = 2) state.
Here,
n1 = 2
n2 = 4
Now, the wavelength `(lamda_1)` will be
`1/lamda_1 = R (1/n_1^2 - 1/n_2^2)`
`R = 1.097 xx 10^7 m^-1`
`1/lamda_1 = 1.097xx 10^7 xx (1/4 - 1/16)`
`rArr 1/lamda_1 = 1.097 xx 10^7 ((4-1)/(16))`
`rArr 1/lamda_1 = (1.097xx10^7xx3)/16`
`rArr lamda_1 = (16xx10^-7)/(3xx1.097)`
= 4.8617 × 10-7
= 486.1 × 10-9
= 487 nm
When an atom makes transition from (n = 4) to (n = 3), the wavelength (λ2) is given by
Here again
`n_1 = 3`
`n_2 = 4`
`1/lamda_2 = 1.097 xx 10^7 (1/9 - 1/16)`
`rArr 1/lamda_2 = 1.097 xx 10^7 ((16 -9)/144)`
`rArr 1/lamda_2 = (1.097xx 10^7 xx 7)/144`
`rArr lamda_2 = 144/(1.097 xx 10^7 xx 7)`
= 1875 nm
Similarly, wavelength (λ2) for the transition from (n = 3) to (n = 2) is given by
When the transition is n1 = 2 to n2 = 3:
`1/lamda_3 = 1.097 xx 10^7 (1/4 - 1/9)`
`rArr 1/lamda_3 = 1.097 xx 10^7 ((9-4)/36) `
`rArr 1/lamda_3 = (1.097xx10^7)/36`
`rArr lamda_3 = (36xx10^7xx5)/36`
`rArr lamda_3 = (36xx10^7)/((1.097)xx5) = 656 nm`
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संबंधित प्रश्न
Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, a thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10−10 m).
(a) Construct a quantity with the dimensions of length from the fundamental constants e, me, and c. Determine its numerical value.
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