Advertisements
Advertisements
Question
Derive an expression for the radius of the nth Bohr orbit for the hydrogen atom.
Advertisements
Solution
Let, me = mass of electron,
−e = charge on electron,
rn = radius of nth Bohr’s orbit,
+e = charge on nucleus,
vn = linear velocity of electron in nth orbit,
Z = number of electrons in an atom,
n = principal quantum number.
∴ The angular momentum = mevnrn
According to second postulate.
mevnrn = `"nh"/("2"pi)` ...(i)
From Bohr’s first postulate,
Centripetal force = Electrostatic force
∴ `("m"_"e""v"_"n"^2)/"r"_"n" = "Ze"^2/(4piepsilon_0"r"_"n"^2)`
∴ `"v"_"n"^2 = "Ze"^2/(4piepsilon_0"r"_"n""m"_"e")` ....(ii)
Squaring equation (i) we get
`"m"_"e"^2 "v"_"n"^2 "r"_"n"^2 = ("n"^2"h"^2)/(4pi^2)`
∴ `"v"_"n"^2 = ("n"^2"h"^2)/(4pi^2"m"_"e"^2"r"_"n"^2)` ….(iii)
Equating equations (ii) and (iii), we get,
`("n"^2"h"^2)/(4pi^2"m"_"e"^2"r"_"n"^2) = "Ze"^2/(4piepsilon_0"r"_"n""m"_"e")`
∴ rn = `("n"^2"h"^2epsilon_0)/(pi"m"_"e""Z"_"e"^2)`
This is the required expression for radius of nth Bohr orbit of the electron.
APPEARS IN
RELATED QUESTIONS
Answer in one sentence:
Name the element that shows the simplest emission spectrum.
The radii of Bohr orbit are directly proportional to ______
For the hydrogen atom, the minimum excitation energy ( of n =2) is ______
If aO is the Bohr radius and n is the principal quantum number then, state the relation for the radius of nth orbit of the electron in terms of Bohr radius and principal quantum number.
Starting with 𝑟 = `(ε_0h^2n^2)/(pimZe^2),` Show that the speed of an electron in nth orbit varies inversely to principal quantum number.
State any two limitations of Bohr’s model for the hydrogen atoms.
Using de Broglie’s hypothesis, obtain the mathematical form of Bohr’s second postulate.
The angular momentum of an electron in the 3rd Bohr orbit of a Hydrogen atom is 3.165 × 10-34 kg m2/s. Calculate Plank’s constant h.
State the postulates of Bohr’s atomic model. Hence show the energy of electrons varies inversely to the square of the principal quantum number.
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.
In hydrogen atom, the de Broglie wavelength of an electron in the first Bohr's orbit is ____________.
[Given that Bohr radius, a0 = 52.9 pm]
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 ____________.
Which of the following models was successful in explaining the observed hydrogen spectrum?
The time of revolution of an electron around a nucleus of charge Ze in nth Bohr orbit is directly proportional to ____________.
Angular speed of an electron in the ground state of hydrogen atom is 4 × 1016 rad/s. What is its angular speed in 4th orbit?
An 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 (h = Planck's constant).
The de-Broglie wavelength of an electron in 4th orbit is ______.
(r = radius of 1st orbit)
Electron in Hydrogen atom first jumps from third excited state to second excited state and then from second excited state to first excited state. The ratio of the wavelengths λ1 : λ2 emitted in the two cases respectively is ______.
If Vn and Vp are orbital velocities in nth and pth orbit respectively, then the ratio Vp: Vn is ______.
An electron of mass m and charge e initially at rest gets accelerated by a constant electric field E. The rate of change of de-Broglie wavelength of this electron at time t ignoring relativistic effects is ______.
The value of Rydberg constant in joule is ______.
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 ______.
Compute the shortest and the longest wavelength in the Lyman series of hydrogen atom.
