Advertisements
Advertisements
प्रश्न
Derive the expression for the energy of an electron in the atom.
Advertisements
उत्तर
Consider an electron orbiting the nucleus of an atom with the atomic number Z in the nth orbit. Let m and -e represent the electron's mass and charge, r the orbit's radius, and v the electron's linear speed.
According to Bohr's first postulate,
centripetal force on the electron = electrostatic force of attraction exerted on the electron by the nucleus
∴ `"mv"^2/"r" = 1/(4piε_0) = "Ze"^2/"r"^2` ...(1)
where ε0 is the permittivity of free space.
∴ Kinetic energy (KE) of the electron
`= 1/2 "mv"^2 = "Ze"^2/(8piε_0"r")` .....(2)
The electric potential due to the nucleus of charge + Ze at a point at a distance r from it is
V = `1/(4piε_0)*"Ze"/"r"`
∴ Potential energy (PE) of the electron
= charge on the electron x electric potential
= `-"e" xx 1/(4piε_0) "Ze"/"r" = - "Ze"^2/(4piε_0"r")` ....(3)
Hence, the total energy of the electron in the nth orbit is
E = KE + PE = `"-Ze"^2/(4piε_0"r") + "Ze"^2/(8piε_0"r")`
∴ E = `"-Ze"^2/(8piε_0"r")` ......(4)
This shows that the total energy of the electron in the nth orbit of the atom is inversely proportional to the radius of the orbit as Z, ε0 and e are constants. The radius of the nth orbit of the electron is
r = `(ε_0"h"^2"n"^2)/(pi"mZe"^2)` ....(5)
where his Planck's constant.
From Eqs. (4) and (5), we get,
`"E"_"n" = - "Ze"^2/(8piε_0)((pi"mZe"^2)/(ε_0"h"^"n"^2)) = - ("mZ"^2"e"^4)/(8ε_0^2"h"^2"n"^2)` ....(6)
This expresses the energy of the electron in the nth Bohr orbit. The minus sign in the expression indicates that the electron is attracted to the nucleus by electrostatic forces.
As m, Z, e, ε0 and h are constant, we get
`"E"_"n" prop 1/"n"^2`
i.e., In a stationary energy state, the energy of the electron is discontinuous and inversely proportional to the square of the primary quantum number.
APPEARS IN
संबंधित प्रश्न
Linear momentum of an electron in Bohr orbit of H-atom (principal quantum number n) is proportional to ______.
According to Bohr's second postulate, the angular momentum of the electron is the integral multiple of `h/(2pi)`. The S.I unit of Plank constant h is the same as ______
What is the energy of an electron in a hydrogen atom for n = ∞?
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.
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.
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.
The radius of electron's second stationary orbit in Bohr's atom is R. The radius of the third orbit will be ______
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?
The total energy of an electron in an atom in an orbit is -3.4 eV. Its kinetic and potential energies are, respectively ______.
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 ____________.
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?
If the speed of an electron of hydrogen atom in the ground state is 2.2 x 106 m/s, then its speed in the third excited state will be ______.
The acceleration of electron in the first orbit of hydrogen atom is ______.
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?
Ratio of centripetal acceleration for an electron revolving in 3rd orbit to 5th orbit of hydrogen atom is ______.
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).
In Bohr model, speed of electron in nth orbit of hydrogen atom is ______. (b = Planck's constant, n = principal quantum number, ∈0 is the permittivity of free space, e = electronic charge)
If a charge on the body is 1 nC, then how many electrons are present on the body?
When an electron in hydrogen atom jumps from third excited state to the ground state, the de-Broglie wavelength associated with the electron becomes ____________.
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 ground state energy of the hydrogen atom is -13.6 eV. The kinetic and potential energy of the electron in the second excited state is respectively ______
The orbital frequency of an electron in the hydrogen atom ______.
In Bohr’s atomic model, speed and time period of revolution of an electron in n = 3 level are respectively.
What is the origin of spectral lines? Obtain an expression for the wave number of a line in hydrogen spectrum.
The radius of the first Bohr orbit in the hydrogen atom is 0.5315 Å. The radius of the second Bohr orbit in the hydrogen atom is ______.
