Advertisements
Advertisements
Question
Obtain an expression for the radius of Bohr orbit for H-atom.
Advertisements
Solution
Let us consider an electron revolving around the nucleus in a circular orbit of radius ‘r’.
According to Bohr’s first postulate, the centripetal force is equal to the electrostatic force of attraction. That is
`"mv"^2/"r"=1/(4piepsilon_o)xx"e"^2/"r"^2`
`"Or,""v"^2="e"^2/(4piepsilon_o"mr")` -------------------(1)
According to the Bohr's second postulate:
`"Angular momentum"= "n""h"/(2pi)`
`"mvr"="n""h"/(2pi)`
Or, `"v"="nh"/(2pi"mr")` -----------------(2)
Or, `"v"^2=("n"^2"h"^2)/(4pi^2"m"^2"r"^2)` ---------------------(3)
Comparing eqn (1) and eqn (3), we get
`"e"^2/(4piepsilon_o"mr")=("n"^2"h"^2)/(4pi^2"m"^2"r"^2)`
`"Or,""r"=(("h"^2epsilon_o)/(pi"me"^2))"n"^2` ----------------------(4)
This equation gives the radius of the nth Bohr orbit.
`"For n"=1,"r"_1=(("h"^2epsilon_o)/(pi"me"^2))=0.537" ---------------(5)"`
`"In general,"" r"_n=(("h"^2epsilon_o)/(pi"me"^2))"n"^2`
The above equation gives the radius of Bohr orbit.
APPEARS IN
RELATED QUESTIONS
The energy associated with the first orbit in the hydrogen atom is - 2.18 × 10-18 J atom-1. What is the energy associated with the fifth orbit?
Draw a neat, labelled energy level diagram for H atom showing the transitions. Explain the series of spectral lines for H atom, whose fixed inner orbit numbers are 3 and 4 respectively.
Lifetimes of the molecules in the excited states are often measured by using pulsed radiation source of duration nearly in the nanosecond range. If the radiation source has a duration of 2 ns and the number of photons emitted during the pulse source is 2.5 × 1015, calculate the energy of the source.
The longest wavelength doublet absorption transition is observed at 589 and 589.6 nm. Calculate the frequency of each transition and energy difference between two excited states.
In accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius 1.5 × 1011 m with orbital speed 3 × 104 m/s. (Mass of earth = 6.0 × 1024 kg)
The gravitational attraction between electron and proton in a hydrogen atom is weaker than the Coulomb attraction by a factor of about 10−40. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.
if `E_p` and `E_k` represent potential energy and kinetic energy respectively, of an orbital electron, then, according to B9hr's theory:
a)`E_k = -E_p"/"2`
b) `E_k = -E_p`
c) `E_k = -2E_p`
d) `E_k = 2E_p`
Using Bohr's postulates, derive the expression for the orbital period of the electron moving in the nth orbit of hydrogen atom ?
Using Bohr's postulates, derive the expression for the total energy of the electron in the stationary states of the hydrogen atom ?
Using Bohr’s postulates, obtain the expression for the total energy of the electron in the stationary states of the hydrogen atom. Hence draw the energy level diagram showing how the line spectra corresponding to Balmer series occur due to transition between energy levels.
A positive ion having just one electron ejects it if a photon of wavelength 228 Å or less is absorbed by it. Identify the ion.
A filter transmits only the radiation of wavelength greater than 440 nm. Radiation from a hydrogen-discharge tube goes through such a filter and is incident on a metal of work function 2.0 eV. Find the stopping potential which can stop the photoelectrons.
Suppose in an imaginary world the angular momentum is quantized to be even integral multiples of h/2π. What is the longest possible wavelength emitted by hydrogen atoms in visible range in such a world according to Bohr's model?
How are various lines of Lyman series formed? Explain on the basis of Bohr’s theory.
Write postulates of Bohr’s Theory of hydrogen atom.
A particle has a mass of 0.002 kg and uncertainty in its velocity is 9.2 × 10−6 m/s, then uncertainty in position is ≥ ____________.
(h = 6.6 × 10−34 J s)
If the radius of first electron orbit in hydrogen atom be r then the radius of the fourth orbit ill be ______.
Calculate the energy and frequency of the radiation emitted when an electron jumps from n = 3 to n = 2 in a hydrogen atom.
Why was a change in the Bohr Model of atom required? Due to which important development (s), concept of movement of an electron in an orbit was replaced by, the concept of probability of finding electron in an orbital? What is the name given to the changed model of atom?
Derive an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n – 1). Also show that for large values of n, this frequency equals to classical frequency of revolution of an electron.
A set of atoms in an excited state decays ______.
An ionised H-molecule consists of an electron and two protons. The protons are separated by a small distance of the order of angstrom. In the ground state ______.
- the electron would not move in circular orbits.
- the energy would be (2)4 times that of a H-atom.
- the electrons, orbit would go around the protons.
- the molecule will soon decay in a proton and a H-atom.
When an electron falls from a higher energy to a lower energy level, the difference in the energies appears in the form of electromagnetic radiation. Why cannot it be emitted as other forms of energy?
The inverse square law in electrostatics is |F| = `e^2/((4πε_0).r^2)` for the force between an electron and a proton. The `(1/r)` dependence of |F| can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. If photons had a mass mp, force would be modified to |F| = `e^2/((4πε_0)r^2) [1/r^2 + λ/r]`, exp (– λr) where λ = mpc/h and h = `h/(2π)`. Estimate the change in the ground state energy of a H-atom if mp were 10-6 times the mass of an electron.
The radius of the innermost electron orbit of a hydrogen atom is 5.3 × 10–11m. The radius of the n = 3 orbit is ______.
The value of angular momentum for He+ ion in the first Bohr orbit is ______.
An electron in H-atom makes a transition from n = 3 to n = 1. The recoil momentum of the H-atom will be ______.
What is the energy of an electron in stationary state corresponding to n = 2?
The total energy of an electron in the nth orbit of the hydrogen atom is proportional to ______.
Specify the transition of an electron in the wavelength of the line in the Bohr model of the hydrogen atom which gives rise to the spectral line of the highest wavelength ______.
