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Show that the first few frequencies of light that is emitted when electrons fall to the n^(th) level from levels higher than n, are approximate harmonics (ie, in the ratio 1:2:3...) when n gt gt 1. |
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Answer» Solution :In an atom of an ELEMENT with atomic no. 2, when an electron makes a transition from `(n + x)^(th)` ORBIT (where x = 1, 2, 3, ....) to `n^(th)` orbit, if FREQUENCY and wavelength of emitted radiation are respectively f and then, `(1)/(lambda)=RZ^(2)((1)/(n^(2))-(1)/((n+x)^(2)))` `:.(f)/(c)=RZ^(2){((n+x)^(2)-n^(2))/((n+x)^(2)(n)^(2))} ( :. c=flambda)` `:.f=RcZ^(2){(n^(2)+2nx+x^(2)-n^(2))/((n+x)^(2)(n^(2)))}` `:.f=RcZ^(2){(2nx+x^(2))/((n+x)^(2)(n^(2)))}` Here `n gt gt 1 and x = 1,2,3.... and so (n+x)= n( :. x lt lt n)`. Also we can neglecte `x^(2)` from the ADDITION. Thus, `r~~RcZ^(2)((2nx)/(n^(4)))` `:.f~~((2rCZ^(2))/(n^(3)))x` `:.f prop x` `:.f_(1):f_(2):f_(3)=x_(1):x_(2):x_(3)` `=1:2:3` |
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