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प्रश्न
Using the expression for the energy of an electron in the nth orbit, Show that `1/lambda = "R"_"H"(1/"n"^2 - 1/"m"^2),` Where symbols have their usual meaning.
उत्तर
- Let, Em = Energy of an electron in mth higher orbit
En = Energy of electron in the nth lower orbit - According to Bohr’s third postulate,
Em - En = hv
∴ v = `("E"_"m" - "E"_"n")/"h"` ….(1) - But Em = `-("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^2"m"^2)` ….(2)
En = `-("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^2"n"^2)` ….(3) - From equations (1), (2) and (3),
v = `(-("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^2"m"^2) - (-("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^2"n"^2)))/"h"`
∴ v = `("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^3")` `[-1/"m"^2 + 1/"n"^2]`
∴ `"c"/lambda = ("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^3")` `[1/"n"^2 - 1/"m"^2]` ......[∵ v = `"c"/lambda`]
where, c = speed of electromagnetic radiation
∴ `1/lambda = ("Z"^2"m"_"e""e"^4)/(8epsilon_0^2"h"^3"c")` `[1/"n"^2 - 1/"m"^2]` - But, `("m"_"e""e"^4)/(8epsilon_0^2"h"^3"c") = "R"_"H"` = Rydberg’s constant
= `1.097 × 10^7 "m"^-1`
∴ `1/lambda = "R"_"H""Z"^2 [1/"n"^2 - 1/"m"^2]` ….(4)
For hydrogen Z = 1,
∴ `1/lambda = "R"_"H"[1/"n"^2 - 1/"m"^2]` ….(5)
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