(a) Draw the energy level diagram for the hydrogen atom. Mark the transitions corresponding to the series lyin

Question

Q. (a) Draw the energy level diagram for the hydrogen atom. Mark the transitions corresponding to the series lying in the ultraviolet region, visible region and infrared region.

(OR)

(b) Draw a diagram to show the variation of binding energy per nucleon with mass number for different nuclei and mention its two features. Why do lighter nuclei usually undergo nuclear fusion?

CBSECBSE Class XII Board 2023Subjective· 3mImportance★★★★★

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Part (a)Concept understanding — Bohr Model Energy Levels

The Intuition: Why Can't an Electron Just Sit Anywhere?

Imagine you're rolling a marble on a staircase. The marble can rest on any step — step 1, step 2, step 3 — but it can never float halfway between two steps. The staircase forces the marble into specific, fixed positions.

That's the core idea of the Bohr model. Before Bohr, physicists thought electrons orbited the nucleus like planets around the sun — they could be at any distance, any energy. But experiments showed something strange: atoms only emit or absorb light at very specific colours (wavelengths), not a continuous rainbow. That meant electrons could only have certain, fixed energies — like the steps of a staircase.

Bohr's genius was to say: an electron in an atom cannot have any arbitrary energy. It can only occupy certain "allowed" energy levels. When it jumps from one level to another, it either absorbs or emits a photon of light whose energy exactly matches the difference between those levels.

The Precise Statement

In the Bohr model of the hydrogen atom (and hydrogen-like ions with one electron), the electron moves in circular orbits around the nucleus. But only those orbits are allowed where the electron's angular momentum is an integer multiple of $\frac{h}{2\pi}$ (where $h$ is Planck's constant).

This quantisation condition leads to a simple formula for the energy of the electron in the $n$ -th orbit:

E_n = -\frac{13.6 \text{ eV}}{n^2}

Here:

$n$ is the principal quantum number — a positive integer ( $n = 1, 2, 3, \dots$ )
$E_n$ is the energy of the electron in that level (in electronvolts)
The negative sign means the electron is bound to the nucleus — you need to add energy to free it

The lowest energy state ( $n=1$ ) is called the ground state. Its energy is $-13.6 \text{ eV}$ . The higher states ( $n=2, 3, 4, \dots$ ) are excited states — they have less negative (higher) energies.

Note

As $n$ increases, the energy levels get closer together. At $n = \infty$ , the energy becomes $0 \text{ eV}$ — the electron is completely free from the atom (ionisation).

What This Explains

When an electron jumps from a higher level ( $n_i$ ) to a lower level ( $n_f$ ), it emits a photon of energy:

\Delta E = E_{n_f} - E_{n_i} = 13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \text{ eV}

This single formula predicts all the spectral lines of hydrogen — the Lyman series (jumps to $n=1$ ), Balmer series (to $n=2$ ), Paschen series (to $n=3$ ), and so on. Each series corresponds to a different "final step" on the staircase. …

Why this formula?

Why the Bohr Model Gives Those Energy Levels

The Bohr model is a beautiful piece of physics because it takes a simple, almost desperate idea — "electrons only exist in certain orbits" — and derives the entire hydrogen spectrum from it. The key is that Bohr didn't just assume the energy levels; he forced them to be consistent with classical physics in one specific way, then broke with it in another.

The Two Non-Negotiable Pieces

First, the electron moves in a circle around the proton. That's pure classical mechanics: the Coulomb attraction provides the centripetal force.

\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r^2} = \frac{mv^2}{r}

This gives you a relation between speed $v$ and radius $r$ :

v^2 = \frac{e^2}{4\pi\varepsilon_0 m r}

Second, the total energy of the electron is the sum of its kinetic and potential energies. Potential energy for a Coulomb force is $-\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}$ (negative because the force is attractive, and we set zero at infinity).

E = \frac12 mv^2 - \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}

Substitute $v^2$ from above:

E = \frac12 \left( \frac{e^2}{4\pi\varepsilon_0 r} \right) - \frac{e^2}{4\pi\varepsilon_0 r} = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0 r}

So far, nothing is quantised. Any radius $r$ gives a valid classical orbit, and the energy just follows from that radius. The problem is that a classical electron in a curved path radiates energy and spirals into the nucleus — atoms should collapse. Bohr needed a rule to pick out stable orbits.

The Quantisation Condition

Bohr's revolutionary step was to postulate that the angular momentum of the electron is quantised in units of $\hbar = h/2\pi$ :

m v r = n \hbar, \quad n = 1, 2, 3, \dots

Why this particular rule? Bohr later said it was the simplest way to get the right answer. But there's a deeper physical motivation: if you think of the electron as a wave (de Broglie's idea, which came a decade later), the condition that a standing wave fits exactly around the circumference $2\pi r = n \lambda$ gives $m v r = n \hbar$ directly. So the quantisation condition is really a wave condition imposed on a particle picture.

Note

The angular momentum quantisation is the only non-classical assumption in the Bohr model. Everything else follows from classical mechanics and electromagnetism.

Deriving the Allowed Radii and Energies

From $m v r = n \hbar$ , we get $v = n\hbar / (mr)$ . Substitute this into the centripetal force equation:

\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r^2} = \frac{m}{r} \left( \frac{n\hbar}{mr} \right)^2 = \frac{n^2 \hbar^2}{m r^3}

Solve for $r$ :

r_n = \frac{4\pi\varepsilon_0 \hbar^2}{m e^2} \, n^2

The constant in front is the Bohr radius $a_0 \approx 0.529 \, \text{Å}$ . So the radii are $r_n = a_0 n^2$ .

Now plug $r_n$ back into the energy expression $E = -\frac12 \frac{e^2}{4\pi\varepsilon_0 r}$ :

E_n = -\frac12 \frac{e^2}{4\pi\varepsilon_0} \cdot \frac{1}{a_0 n^2} = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0 a_0} \cdot \frac{1}{n^2}

Substitute $a_0 = \frac{4\pi\varepsilon_0 \hbar^2}{m e^2}$ :

E_n = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0} \cdot \frac{m e^2}{4\pi\varepsilon_0 \hbar^2} \cdot \frac{1}{n^2} = -\frac{m e^4}{8 \varepsilon_0^2 h^2} \cdot \frac{1}{n^2}

E_n = -\frac{13.6 \text{ eV}}{n^2}

Part (b)Concept understanding — Nuclear Binding Energy per Nucleon

The Intuition: Why Are Nuclei Stuck Together?

Imagine a nucleus as a tight cluster of protons and neutrons. Protons all carry positive charge, so they should be violently repelling each other. Yet the nucleus holds together. That means there must be an even stronger attractive force — the strong nuclear force — acting between nucleons (protons and neutrons). But here's the catch: this force is extremely short-range. A nucleon only feels the pull from its immediate neighbours, not from nucleons far across the nucleus.

So the nucleus is a tug-of-war. The strong force pulls nucleons together, but the electrostatic repulsion between protons tries to blow the nucleus apart. For a nucleus to be stable, the strong force must win.

Now, if you want to break a nucleus apart into its individual protons and neutrons, you have to do work against the strong force — you have to supply energy. That energy, once supplied, is stored in the separated nucleons as extra mass. This is the core idea: the mass of a stable nucleus is always less than the sum of the masses of its individual protons and neutrons. The missing mass is called the mass defect, and the energy equivalent of that missing mass (via $E = mc^2$ ) is the binding energy.

Note

Binding energy is the energy you must put in to completely separate a nucleus into its constituent nucleons. It is not energy stored inside the nucleus like fuel; it is the energy that holds the nucleus together.

The Precise Statement

For a nucleus with $Z$ protons, $N$ neutrons, and mass $m_{\text{nucleus}}$ :

\text{Mass defect } \Delta m = Z m_p + N m_n - m_{\text{nucleus}}

where $m_p$ and $m_n$ are the masses of a free proton and neutron. Then the total binding energy is:

\text{BE} = \Delta m \, c^2

But a big nucleus has more nucleons, so its total binding energy will naturally be larger. To compare how tightly bound different nuclei are, we use binding energy per nucleon:

\text{BE/A} = \frac{\text{BE}}{A}

where $A = Z + N$ is the mass number. This is the average energy you'd need to remove one nucleon from the nucleus. A higher BE/A means a more stable nucleus.

The Famous Curve: Why Iron is Special

If you plot BE/A against mass number $A$ , you get a curve that rises steeply for light nuclei, peaks at iron-56 (about 8.8 MeV per nucleon), and then slowly falls for heavier nuclei.

Important

Iron-56 has the highest binding energy per nucleon of any nuclide. It is the most stable nucleus in nature.

This shape tells you two things:

1. Fusion of light nuclei releases energy. If you take two very light nuclei (like hydrogen isotopes) and smash them together to form a medium-mass nucleus, the product has a higher BE/A than the reactants. The difference in binding energy is released as kinetic energy of the products. This is how stars burn.

2. Fission of heavy nuclei releases energy. If you split a very heavy nucleus (like uranium-235) into two medium-mass fragments, those fragments also have higher BE/A than the original. Again, the difference comes out as energy. This is how nuclear reactors work. …

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Part (a): the hydrogen energy-level diagram has $E_n=-13.6/n^2\,\text{eV}$ ; Lyman ( $\to n=1$ ) is UV, Balmer ( $\to n=2$ ) is visible, Paschen ( $\to n=3$ ) is infrared. Part (b): the binding-energy-per-nucleon curve peaks near iron; its features are the iron maximum and the low values at both extremes, and lighter nuclei fuse because the product has higher B.E./A.

Energy level diagram of the hydrogen atom showing the discrete energy levels n=1 (-13.6 eV) through n=5 (-0.54 eV) and the continuum above E=0, with downward transition arrows grouped into the Lyman series (transitions ending at n=1, ultraviolet region), the Balmer series (ending at n=2, visible region), and the Paschen series (ending at n=3, infrared region).

Part (a)

Energy-level diagram and spectral series

The allowed energies of hydrogen are

E_n=-\frac{13.6}{n^2}\,\text{eV},\qquad n=1,2,3,\dots

On the diagram the levels are horizontal lines: $n=1$ at $-13.6\,\text{eV}$ (ground state) at the bottom, $n=2$ at $-3.4\,\text{eV}$ , $n=3$ at $-1.51\,\text{eV}$ , and so on, bunching up toward $E=0$ (ionisation) as $n\to\infty$ . A downward arrow between two levels is a photon emission of energy $h\nu=E_{n_2}-E_{n_1}$ .

Marking the series:

Lyman series — ultraviolet: all transitions that end on $n=1$ (from $n=2,3,4,\dots$ ). Largest energy gaps $\Rightarrow$ shortest wavelengths $\Rightarrow$ UV.
Balmer series — visible: transitions that end on $n=2$ (from $n=3,4,5,\dots$ ). These fall in the visible band. …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 85 on this concept.

CBSE 2026Set 55/1/11 markMCQ
$Q. For questions 13 to 16, two statements are given - one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) below: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Both Assertion (A) and Reason (R) are false. Assertion (A) : The mass of a nucleus is less than the sum of the masses of the constituent nucleons. Reason (R) : Energy is absorbed when the nucleons are bound together to form a nucleus.$
›Reveal solutionSolution
The key idea is mass defect: the mass of a stable nucleus is always less than the sum of its individual nucleon masses because the binding energy released during formation reduces the total mass. The Assertion is true, but the Reason is false — energy is released, not absorbed, when nucleons bind.

This question tests your understanding of mass defect and binding energy — two of the most beautiful consequences of Einstein’s mass-energy equivalence, $E = mc^2$ .

When nucleons (protons and neutrons) come together to form a nucleus, they attract each other via the strong nuclear force. To pull them apart into isolated nucleons, you must supply energy — that energy is called the binding energy. Conversely, when they bind, that same amount of energy is released into the surroundings (usually as gamma rays).

Here’s the crucial link: because energy is released, the system loses mass. The mass of the bound nucleus is less than the sum of the masses of its individual nucleons. This missing mass, multiplied by $c^2$ , exactly equals the binding energy. That’s the mass defect.

Now let’s examine each statement carefully.

Assertion (A): “The mass of a nucleus is less than the sum of the masses of the constituent nucleons.”

This is a well-established experimental fact for every stable nucleus. For example, a helium-4 nucleus has mass $4.0026\ \text{u}$ , but two protons and two neutrons add up to $4.0330\ \text{u}$ . The difference ( $0.0304\ \text{u}$ ) is the mass defect. So Assertion (A) is true.

Reason (R): “Energy is absorbed when the nucleons are bound together to form a nucleus.” …
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CBSE 2026Set 55/1/11 markMCQ
$Q. For questions 13 to 16, two statements are given - one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) below: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Both Assertion (A) and Reason (R) are false. Assertion (A) : In Bohr model of hydrogen atom, the energy levels are discrete and quantised. Reason (R) : In a hydrogen atom, the electrostatic force on the electron provides the necessary centripetal force to it to revolve around the nucleus.$
›Reveal solutionSolution

The key idea is that Bohr’s model indeed has discrete energy levels (true), and the reason given — that electrostatic force provides centripetal force — is also true, but it is a classical condition that does not explain quantisation. So both are true, but the reason is not the correct explanation of the assertion. The answer is (B).

Let’s unpack this carefully. The question is from the Assertion-Reason format, so we need to check two things independently: whether each statement is true, and then whether the reason correctly explains the assertion.

1. Check the Assertion (A):

“In Bohr model of hydrogen atom, the energy levels are discrete and quantised.”

This is absolutely true. Bohr’s model was revolutionary precisely because it introduced quantised orbits — electrons can only occupy certain allowed energy levels, and they jump between these levels by absorbing or emitting photons of specific energies. The energy of the $n$ -th level is given by $E_n = -\frac{13.6}{n^2} \text{ eV}$ , which is clearly discrete. So Assertion (A) is true.

2. Check the Reason (R):

“In a hydrogen atom, the electrostatic force on the electron provides the necessary centripetal force to it to revolve around the nucleus.”

This is also true — it’s a standard Newtonian condition for any circular orbit. For an electron of mass $m$ and speed $v$ at a distance $r$ from the nucleus, the Coulomb attraction $\frac{ke^2}{r^2}$ equals the centripetal force $m v^2 / r$ . So Reason (R) is true.

3. Now the crucial part: Does (R) correctly explain (A)?

The reason describes a classical force balance — it’s the same condition that would hold for a planet orbiting the Sun. But that classical condition alone does not lead to discrete, quantised energy levels. In fact, classical physics would allow any orbit radius, and hence any energy. Bohr had to add an extra postulate — quantisation of angular momentum ( $m v r = n \hbar$ ) — to get discrete levels. The electrostatic-centripetal balance is necessary for the model, but it is not the reason for quantisation. So (R) is true but does not explain (A). …
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CBSE 2026Set 55/2/11 markMCQ
$n = 5$
›Reveal solutionSolution
The energy difference between levels determines photon wavelength through $E = \frac{hc}{\lambda}$ . Since the $5 \to 2$ transition releases less energy than $5 \to 1$ , its photon has a longer wavelength. The answer is $\boxed{\frac{32}{7}\lambda}$ — option (D).

The Bohr model tells us that when an electron drops from a higher energy level to a lower one, it emits a photon whose energy exactly equals the energy difference between those levels. The key relationship is $E_{\text{photon}} = \frac{hc}{\lambda}$ , which shows that energy and wavelength are inversely related: a smaller energy gap produces a longer wavelength.

In hydrogen, the energy of the $n$ -th level is given by:
$E_n = -\frac{13.6 \text{ eV}}{n^2}$
The negative sign indicates that the electron is bound to the nucleus. When the electron transitions from level $n_i$ to $n_f$ , the energy released is:
$\Delta E = E_{n_i} - E_{n_f} = 13.6 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \text{ eV}$
This energy becomes the photon's energy: $\Delta E = \frac{hc}{\lambda}$ .

For the first transition ( $5 \to 1$ ):

Calculate the energy difference:

$\Delta E_1 = 13.6 \left(\frac{1}{1^2} - \frac{1}{5^2}\right) = 13.6 \left(1 - \frac{1}{25}\right) = 13.6 \times \frac{24}{25}$

This energy corresponds to wavelength $\lambda$ :

$\frac{hc}{\lambda} = 13.6 \times \frac{24}{25}$
For the second transition ( $5 \to 2$ ):

Calculate the energy difference:

$\Delta E_2 = 13.6 \left(\frac{1}{2^2} - \frac{1}{5^2}\right) = 13.6 \left(\frac{1}{4} - \frac{1}{25}\right)$

Find a common denominator:

$\Delta E_2 = 13.6 \left(\frac{25 - 4}{100}\right) = 13.6 \times \frac{21}{100}$

This energy corresponds to wavelength $\lambda'$ : …
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CBSE 2026Set V11 markMCQ
$Q. Let K be the kinetic energy, U be the potential energy and E be the total energy of an electron revolving around the nucleus in a hydrogen atom, then which of the following is correct?$
(a) $K > 0,\ U > 0,\ E > 0$
(b) $K > 0,\ U < 0,\ E < 0$
(c) $K > 0,\ U > 0,\ E < 0$
(d) $K < 0,\ U < 0,\ E < 0$
›Reveal solutionSolution

Option (b) $K > 0,\ U < 0,\ E < 0$ . …

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CBSE 2026Set ANNUAL1 markMCQ
$Q. Formula for total energy of the electron in the nth stationary state of the hydrogen atom is -$
(i) $-\dfrac{11.2}{n^2}$ eV
(ii) $+\dfrac{11.2}{n^2}$ eV
(iii) $-\dfrac{13.6}{n^2}$ eV
(iv) $+\dfrac{13.6}{n^2}$ eV
›Reveal solutionSolution

Total energy of the electron in the nth Bohr orbit of hydrogen is $-13.6/n^2$ eV.

…

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CBSE 2026Set A1 markMCQ
$Q. The solar spectrum is (A) continuous (B) line spectrum (C) spectrum of black lines (D) spectrum of black bands$
›Reveal solutionSolution

The solar spectrum is a line-absorption spectrum: a continuous background crossed by dark (Fraunhofer) lines.

The hot, dense interior of the Sun emits a continuous spectrum. As this light passes through the cooler gases of the Sun's outer atmosphere, atoms there absorb their characteristic wavelengths, leaving dark lines (Fraunhofer lines) superimposed on the …

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CBSE 2026Set A1 markMCQ
$Q. The kinetic energy (K) of an electron in a Bohr orbit is related to its potential energy (U) by (A) K = U (B) K = -U (C) K = -U/2 (D) K = -2U$
›Reveal solutionSolution

For an electron in a Bohr orbit, K = -U/2 (with U taken negative), which also gives total energy E = -K.

In a hydrogen-like Bohr orbit the Coulomb attraction provides the centripetal force:

$\frac{k e^2}{r^2} = \frac{m v^2}{r} \;\Rightarrow\; m v^2 = \frac{k e^2}{r}.$

So the kinetic energy is

$K = \tfrac{1}{2} m v^2 = \frac{k e^2}{2r}.$ …

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CBSE 2026Set ANNUAL1 mark
$Q. Write the definition of emission line spectrum.$
›Reveal solutionSolution

When atoms of a rarefied gas are excited (e.g. by heating or an electric discharge) and their electrons fall back to lower energy levels, they emit light only at specific wavelengths, producing a spectrum of separated bright lines rather than a continuous band.

An emission line spectrum is obtained when the light emitted directly by a source of excited atoms (such as a gas discharge tube) is passed through a spectrometer/prism. Because each element's electrons can only occupy discrete (quantised) energy levels, transitions between these levels emit photons of only certain specific energies (and hence specific wavelengths/frequencies). The resulting spectrum consists of a series of bright, sharp coloured lines at these part …

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CBSE 2026Set ANNUAL1 mark
$Q. If the binding energy of oxygen nucleus 16/8 O is 128 MeV, then calculate the binding energy per nucleon.$
›Reveal solutionSolution

Binding energy per nucleon is simply the total binding energy divided by the number of nucleons (mass number A).

For 16/8 O, mass number A = 16 (8 protons + 8 neutrons), and total binding energy = 128 MeV (given). …

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CBSE 2026Set ANNUAL1 markMCQ
$Q. Case study: Bohr's model addressed the instability of the Rutherford model by introducing quantization. The model is based on three postulates, which successfully explained the discrete line spectrum of hydrogen. According to the model, the radius of the nth stationary orbit is r_n \propto n², and the total energy is E_n = -13.6 eV / n², when an electron jumps from a higher energy level (E_i) to a lower one (E_f), a photon of energy hν = E_i - E_f is emitted. Transitions ending at the n = 1 level form the Lyman series. The energy of an electron in the first excited state (n = 2) of hydrogen is:$
(a) –13.6 eV
(b) –6.8 eV
(c) –3.4 eV
(d) –1.7 eV
›Reveal solutionSolution

Bohr energy levels of hydrogen follow $E_n = -13.6/n^2$ eV; for $n=2$ , $E_2 = -3.4$ eV.

$E_n = \dfrac{-13.6}{n^2}\,\text{eV}$

For the first excited state, $n=2$ : …

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CBSE 2026Set ANNUAL1 markMCQ
$Q. Case study: Bohr's model addressed the instability of the Rutherford model by introducing quantization. The model is based on three postulates, which successfully explained the discrete line spectrum of hydrogen. According to the model, the radius of the nth stationary orbit is r_n \propto n², and the total energy is E_n = -13.6 eV / n², when an electron jumps from a higher energy level (E_i) to a lower one (E_f), a photon of energy hν = E_i - E_f is emitted. Transitions ending at the n = 1 level form the Lyman series. Transitions ending at n = 3 belong to which series?$
(a) Lyman series
(b) Balmer series
(c) Paschen series
(d) Brackett series
›Reveal solutionSolution

Transitions of hydrogen electrons ending on the $n=3$ level constitute the Paschen series (infrared).

…

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CBSE 2026Set ANNUAL1 mark
$Q. Case study: Bohr's model addressed the instability of the Rutherford model by introducing quantization. The model is based on three postulates, which successfully explained the discrete line spectrum of hydrogen. According to the model, the radius of the nth stationary orbit is r_n \propto n², and the total energy is E_n = -13.6 eV / n², when an electron jumps from a higher energy level (E_i) to a lower one (E_f), a photon of energy hν = E_i - E_f is emitted. Transitions ending at the n = 1 level form the Lyman series. Explain the physical significance of the negative sign in the expression for the total energy E_n of an electron in the Bohr model. OR What is the ratio of the radius of the third Bohr orbit (n = 3) to the radius of the first Bohr orbit (n = 1)?$
›Reveal solutionSolution

A negative total energy means the electron is in a bound state — energy must be added to remove it to infinity (where E=0).

The total energy of an electron in the $n^{th}$ Bohr orbit is $E_n = -\dfrac{13.6}{n^2}\,\text{eV}$ . The negative sign signifies that the electron is bound to the nucleus by the attractive electrostatic force — its potential energy (taken as zero at infinite separation) is more negative than its kinetic energy is positive, so the net energy is negative. This means external energy (at least $|E_n|$ ) must be supplied …

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Q.(a) Draw the energy level diagram for the hydrogen atom. Mark the transitions corresponding to the series lying in the ultraviolet region, visible region and infrared region.

Part (a)

Energy-level diagram and spectral series

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