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Energy Levels are the discrete energy values absorbed by a quantum mechanical system or particle that is bound or spatially restricted. On the other hand, conventional particles may accept any quantity of energy. Energy Levels are mainly the levels of energy of electrons confined by the electromagnetic current of the nucleus in atoms, ions, or molecules. It can also refer to the energy state of nuclei or vibration energy levels in molecules.
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Key Terms: Energy Levels, energy level equation, Energy Level Formula, Electromagnetic current, Quantum mechanics
Energy Level Formula
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The Bohr Model of the hydrogen atom should be taken into account when calculating the rotational energy level formula.
- In the standard model of the hydrogen atom, an assumption was made concerning the quantization of atoms.
- According to this theory, electrons orbit the nucleus in orbits or shells with a fixed radius.
- Only shells with a radius equal to the Energy Level Formula were permitted.
- Additionally, no electrons can be present between the shells.
- The mathematical expression for the permitted value of the atomic radius in the energy level is,
r(n) = n2 + r(1)
As a result, the equation for energy state for radius is often known as Bohr's formula.
Energy Levels
Read More:
| Energy Level Formula – Related Topics | ||
|---|---|---|
| Alpha Particle Mass | Fermi Energy | Spectral Series |
| Modern Physics | Value of c | Ionizing Radiation |
Energy of Electron Formula
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To compute the energies of electrons at the nth level of the hydrogen atom, Bohr utilized electrons in circular and quantized orbits. This may be observed in the electron energy level formula, which is as shown below.
E(n)= \({-1 \over n_2}\) × 13.6eV
- A hydrogen electron's least possible energy constant value is 13.6 eV.
- The energy in photons is absorbed by an electron, allowing it to be stimulated to a higher energy state.
- The excited electron becomes less stable after changing to a higher energy state, also known as the excited state, and will release a photon to revert to a low and much more stable level of energy.
- The discrepancy in energy between the two-energy states equals the produced energy for a certain transition.
The energy can be computed using the below energy level formula.
hv = ΔE = \(({1 \over n^2_{low}} - {1 \over n^2_{high}})13.6eV\)
Energy Level Formula – Solved ExampleQues. In terms of electrons absorbing and releasing photons to alter energy levels, the hydrogen spectrum has the greatest wavelength (nhigh) of 4 and the shortest wavelength (nlow) of 2. Using the Bohr energy level formula, calculate the energy. [Click here for Solution] Ans. Given, nhigh = 4 nlow = 2 By using the Bohr formula energy levels, we get, hv = ΔE = (1 / n2 low – 1 / n2 high) x 13.6eV ΔE = (1 / 22 – 1 / 42) x 13.6eV ΔE = (0.25 - 0.0625) x 13.6eV ΔE = 0.1875 x 13.6eV ΔE = 2.55eV |
The equation for the energy level formula of hydrogen atoms is
E = \({E_0 \over n^2}\)
where Eo is 13.6 eV and n is any natural number from 1, 2……and so on
Hydrogen Energies and Spectrum
Also Read:
Things to Remember
- The Value of the Atomic Radius – r(n) = n2 x r(1)
- Energy of atom in the nth level of the hydrogen atom – E(n)= \({-1 \over n_2}\) × 13.6eV
- The value of energy emitted for a specific transition – hv = ΔE = \(({1 \over n^2_{low}} - {1 \over n^2_{high}})13.6eV\)
- Energy Level Formula – E = \({E_0 \over n^2}\)
Previous Year Questions
- Electron in hydrogen atom first jumps from third excited state to second excited… [NEET 2012]
- One standard atomic mass unit is… [JIPMER 1998]
- Polar molecules are the molecules… [NEET 2021]
- Alpha particle consists of… [NEET 2019]
- A sample of radioactive element has a mass of 10g at… [NEET 2003]
- When an electron does transition from n=4 to n = 2… [NEET 2000]
- When hydrogen atom is in its first excited level, its radius, is… [NEET 1997]
- For which one of the following, Bohr model is not valid… [NEET 2020]
- Curie is a unit of… [NEET 1989]
- An electron of a stationary hydrogen atom passes from the fifth energy… [NEET 2012]
- An electron in the hydrogen atom jumps from excited state n to… [NEET 2011]
- If the energy of a hydrogen atom in nth orbit is En… [NEET 2001]
- Hydrogen atom in ground state is excited by a monochromatic radiation… [NEET 2014]
- Given the value of Rydberg constant is 107 m… [NEET 2016]
- Energy is released in nuclear fission is due to… [NEET 2001]
- Due to earth's magnetic field, the charged cosmic rays particles… [NEET 1997]
- In terms of Bohr radius a0, the radius of the second Bohr orbit… [NEET 1992]
- In any fission process the ratio mass of fission products… [NEET 2005]
- In a Rutherford scattering experiment when a projectile of charge… [NEET 2009]
- In the spectrum of hydrogen, the ratio of the longest wavelength… [NEET 2015]
Sample Questions
Ques. What is Bohr Formula Energy Levels? (3 marks)
Ans. Bohr's hydrogen model is based on the nonclassical idea that electrons move in certain shells, or orbits, around the nucleus. Using Bohr's model, the following energies for an electron in the shell were calculated: n:
E(n) = -1 / n2 x 13.6 eV
Bohr characterised the hydrogen spectrum as electrons absorbing and releasing photons to change energy levels, with photon energy measured in eV.
hv = ΔE = (1 / n2low – 1 / n2high) x 13.6 eV
Bohr's concept didn't work for systems with more than one electron.
Ques. Explain the energy level transition equation. (3 marks)
Ans. An electron gets absorbed energy in the form of light energy throughout the transformation. The change in energy E of the electron during the transition is equal to the energy change E of the photon E absorbed / E released. We can compute the frequency and wavelength of the following formula using DeBroglie wave characteristics.
E = hv = hcλ
Planck's constant is 6.63 x 10-34 J's, v signifies frequency, w denotes wavelength, and c = 3.00 x 108 m/s denotes the speed of light.
Ques. Sketch up a diagram of the Geiger-Marsden experiment. How did the scattering of particles by a thin gold foil give an essential method for determining a nucleus size upper limit? Briefly explain. (5 marks)
Ans. The schematic arrangement is depicted below:
Geiger Marsden Experiment
The working is as follows:
A beam of -particles deviates at different angles, with varying probabilities.
Particles with the lowest impact parameter experience more scattering – rebounding when they collide head-on.
When the impact parameter increases, the particle is nearly undeviated.
Because the number of incident particles rebounding is a tiny proportion of the total, the number of -particles colliding head-on is negligible. This means that the atom's whole positive charge is concentrated in a small area. This proves that the nucleus of an atom has a maximum size.
Ques. Compute (5 marks)
(a) Using Bohr's theory of the hydrogen atom, calculate the total energy of the electron in the atom's stationary states.
(b) How would the radius and ground state energy of a muonic atom be modified if the electron in the atom was replaced by a particle (muon) with the same charge but a mass around 200 times that of the electron?
(c) Find the wavelength of the first spectral line in this atom's corresponding Lyman series.
Ans. The calculation is as follows:
(a) Total energy of the electron in Bohr's stationary orbit —K. E owing to velocity and P.E related to electron position.
mv2 / r = K Z e2 / r2
½ m v2 = ½ K Z e2 / r
i.e. K.E of electron ½ m v2 = ½ K Z e2 / 2r …(i)
potential due to the nucleus = K Ze / r
P.E of electron = potential x charge
= K Z e (-e) / r
= -K Z e2 / r …(ii)
P.E of electron in the orbit, E = KE + PE
= K Z e2 / 2r + (= -K Z e2 / r)
= (½ K Z e2 / 2r) – (K Z e2 / r)
= -K Z e2 / 2r)
Putting the value of r = n2 h2 / 4 π. 2 m K z e2
We get
E = (-K Z e2 / 2) x (4 π.2 m K z e2 / n2 h2)
= - (2 π. 2 m K2 Z2 e4 / 2) / (n2 h2)
(b) Radius becomes,
r’ = (n2 h2) / (4 π.2 M’ K z e2)
= (n2 h2) / (4 π.2 200m K z e2)
= (1 / 200) x r
(c) Since = R(1 / (12 – 22))
= 1.097 x 107 (1/1 – ¼)
= 1.097 x 107 ((4-1) / 4)
= 1.097 x 107 (3/4)
= 0.822 x 107
Hence, wavelength,
λ = 1 / V = 4 / 3R
= 4 / (3 x 1.097 x 107)
= 1.215 x 10-7
= 1215 x 10-10 m
= 1215 A
Ques. (a) Identify two major flaws in the Rutherford model that prevent it from explaining the observable properties of atomic spectra. What role did the hydrogen atom have in Bohr's model? Using the Rydberg formula, determine the wavelength of the H line.
(Assume R = 1.1 107 m-1)
(b) Using Bohr's postulates, calculate the radius of the hydrogen atom's nth orbit. (5 marks)
Ans. (a) Rutherford Model Limitations:
Electrons in a circular orbit around the nucleus would accelerate as their energy was spent, spiralling into the nucleus.
It must be able to create a broad spectrum of light.
Explanation based on Bohr's hydrogen atom model:
The electrons of an atom can revolve in stable orbits without producing radiant energy.
Energy is released or absorbed only when an electron jumps from one stable orbit to another stable orbit. As a result, a unique spectrum appears.
When an electron moves from the nf = 3 to the ni = 2 orbit, the H line appears. It's from the Balmer collection.
1 / λ = R(1/22 – 1/32)
1 / λ = 1.1 x 107(1/4 – 1/9)
λ = 656.3nm
(b) Radius of nth orbit is
We have, mv2 / rn
From the Bohr’s postulates
m vn rn = nh / 2π
vn = nh / 2π m rn
Ques. Using Bohr's atomic model postulates, get the equation for the radius of the nth electron orbit. As a consequence, the formula for Bohr's radius may be found. (5 marks)
Ans. The essential postulates of Bohr's atomic model are as follows:
Every atom has a nucleus, a central core that contains all of the atom's positive charge and mass. There are enough electrons in a circular orbit around the nucleus. The centripetal force required for revolution is provided by the electrostatic attraction between the electron and the nucleus.
The electron can only resolve in stationary orbits, which are distinct non-radiating orbits. The total angular momentum of a revolving electron is given as an integral multiple of h/2, h is the plank constant. mvr = nh / 2π
Energy is only released when an electron leaps from one allowed orbit to another. The difference in total energy of the two permissible orbits is absorbed when an electron jumps from the inner to outer orbit, and when an electron jumps from the outer to inner orbit, it is ejected.
The radii of Bohr's stationary orbits are equal to the angular momentum of an electron for any allowed orbit, according to Bohr's postulates.
mvr = nh / 2π
or
v = nh / 2πmr …(i)
The centripetal force is also comparable to the electrostatic force between the electron and the nucleus, according to Bohr's postulates.
Mv2 / r = K(Ze . e) / r2
Or
(m/r) x (n2 h2 / 4 π. 2 m2 r2) = K Z e2 / r2
or
r 4 π.2 m K Z e2 = n2 h2
r = n2 h2 / 4 π. 2 m K Z e2 …(ii)
r = h2 / 4 π m K e2
It is the expression for Bohr’s radius, which is about 0.5 A, for first orbit of hydrogen atom, n = 1, Z = 1, then we have r = 0.529 A
Ques. Define the term ionisation energy. When an electron in a hydrogen atom is replaced with a particle with a mass 200 times that of the electron but the same charge, how does the ionisation energy change? (3 marks)
Ans. Ionisation energy is defined as the amount of ionisation that occurs during the process of ionisation. The minimum energy required to free an electron from the hydrogen atom's ground state is called energy. The following formula is used to compute the ionisation energy:
Eo = (m e2) / (8µo2 h2) i.e. Eo ∝ m
The ionisation energy of a particular particle will increase by 200 times, and the mass of that particle will increase by 200 times.
Ques. The hydrogen atom's ground state energy is -13.6 eV, and its Bohr radius is 0.53. (5 marks)
a) Calculate the energy necessary to transport an electron from one excited state to the next.
b) Find the kinetic energy and the atom's orbital radius in its second excited state.
Ans. a) En = Eo / n2
E3 = -13.6 / (3)2
= - 13.6 / 9
= -1.51 eV
b) KE = - (total energy) = -(-1.51)
=1.51 eV
rn = ro x n2
r3 = ro x (3)2
= (0.53) x 9
= 4.77 A
Ques. The energy of a hydrogen atom in its ground state is -13.6 eV. (5 marks)
a) Calculate the energy necessary to transfer an electron from the ground state to the atom's first excited state.
b) Calculate the kinetic energy and orbital radius of the atom in its initial excited state. (With a Bohr radius of 0.53, this is the case.)
Ans. a) En ∝ 1/n2
Energy of the first excited state
= E1 / 22 = -13.6 / 4
= -3.4 eV
Energy required = [(-3.4) - (-13.6)]
= 10.2 eV
b) Kinetic Energy of the first excited state
= -energy of first excited state
= - (-3.4) = 3.4 eV
Orbital radius of the first excited state
rn ∝ n2
n = 2
rn = ron2
= (0.53) x (2)2
= 2.12 A
Ques. The Bohr radius of a hydrogen atom in its ground state is 5.3 x 10-11 m. The atom is stimulated to the point where its radius is 21.2 x 10-11 m. Determine (5 marks)
a) The major quantum number's value and
b) The atom's total energy is in this excited state.
Ans. Given,
a) r1 = 5.3 x 10-11 m
r2 = 21.2 x 10-11 m
r2 / r1 = (21.2 x 10-11) / (5.3 x 10-11)
= 4
Since r ∝ n2
(n2 / n1)2 = r2 / r1 =4
n2 / n1 = √4 = 2
n2 = 2n1
As a result, the primary quantum number in this excited state has a value of 2.
b) Electron ground state energy = -13.6 eV
En = – 13.6 / n2
En = – 13.6 / (2)2
= – 13.6 / 4
= – 3.4 eV
Total energy of the atom in this excited state is
= – 3.4 eV
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