Advertisements
Advertisements
Question
Obtain an expression for wavenumber, when an electron jumps from a higher energy orbit to a lower energy orbit. Hence show that the shortest wavelength for the Balmar series is 4/RH.
Advertisements
Solution
Expression for wavenumber:
- Let, Em = Energy of an electron in mth higher orbit
En = Energy of an electron in an nth lower orbit - According to Bohr’s third postulate,
Em − En = hν
∴ ν = `("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 × 107 m−1
∴ `1/lambda = "R"_"H""Z"^2 [1/"n"^2 - 1/"m"^2]` ….(4)
This is the required expression. - For shortest wavelength for Balmer series:
n = 2 and m = ∞
`1/lambda = "R"_"H"[1/2^2 - 1/∞]`
= `"R"_"H"/4`
∴ `lambda = 4/"R"_"H"`
APPEARS IN
RELATED QUESTIONS
The speed of electron having de Broglie wavelength of 10 -10 m is ______
(me = 9.1 × 10-31 kg, h = 6.63 × 10-34 J-s)
State Bohr's second postulate for the atomic model. Express it in its mathematical form.
State any two limitations of Bohr’s model for the hydrogen atoms.
Using the expression for the radius of orbit for the Hydrogen atom, show that the linear speed varies inversely to the principal quantum number n the angular speed varies inversely to the cube of principal quantum number n.
Which of the following statements about the Bohr model of the hydrogen atom is FALSE?
How many moles of electrons are required for reduction of 9 moles of Cr3+ to Cr?
For which one of the following, Bohr model is not valid?
In the nth orbit, the energy of an electron `"E"_"n"= -13.6/"n"^2"eV"` for hydrogen atom. The energy required to take the electron from first orbit to second orbit will be ____________.
For an electron, discrete energy levels are characterised by ____________.
If m is mass of electron, v its velocity, r the radius of stationary circular orbit around a nucleus with charge Ze, then from Bohr's second postulate, the kinetic energy `"K.E." = 1/2"mv"^2` of the electron in C.G.S. system is 2 equal to ____________.
In hydrogen emission spectrum, for any series, the principal quantum number is n. Corresponding maximum wavelength λ is ______.
(R = Rydberg's constant)
When an electron in hydrogen atom is excited from its 3rd to 5th stationary orbit, tbe change in angular momentum of electron is (Planck's constant: h = 6.62 x 10-34 Js) ____________.
In hydrogen spectnun, the wavelengths of light emited in a series of spectral lines is given by the equation `1/lambda = "R"(1/3^2 - 1/"n"^2)`, where n = 4, 5, 6 .... And 'R' is Rydberg's constant.
Identify the series and wavelenth region.
In Bohr's model of hydrogen atom, which of the following pairs of quantities are quantized?
If a charge on the body is 1 nC, then how many electrons are present on the body?
Using Bohr's quantization condition, what is the rotational energy in the second orbit for a diatomic molecule. (I = moment of inertia of diatomic molecule, h = Planck's constant)
The electron of mass 'm' is rotating in first Bohr orbit of radius 'r' in hydrogen atom. The orbital acceleration of the electron in first orbit is ______.
(b =Planck's constant)
The third line of the Balmer series, in the emission spectrum of the hydrogen atom, is due to the transition from the ______.
When an electron in hydrogen atom revolves in stationary orbit, it ______.
Which of the following series of transition of hydrogen spectrum falls in visible region?
The speed of an electron in ground state energy level is 2.6 × 106 ms-1, then its speed in third excited state will be ______.
Ultraviolet light of wavelength 300 nm and intensity 1.0 Wm−2 falls on the surface of a photosensitive material. If one percent of the incident photons produce photoelectrons, then the number of photoelectrons emitted from an area of 1.0 cm2 of the surface is nearly ______.
Calculate the energy of the electron in the ground state of the hydrogen atom. Express it in joule and in eV.
What is the origin of spectral lines? Obtain an expression for the wave number of a line in hydrogen spectrum.
Compute the shortest and the longest wavelength in the Lyman series of hydrogen atom.
Find the ratio of radius of 1st Bohr orbit to that of 4th Bohr orbit.
