Structure of Atom Class 11 Notes: CBSE Chemistry Chapter 2

VALENCY OF SOME COMMON BASIC AND ACIDIC RADICALS

Electrovalent Positive Ions (Basic Radicals)

ELECTROVALENT NEGATIVE IONS (ACIDIC RADICALS)

1. INTRODUCTION

This chapter delves into the mysterious and surprising world of atoms on the inside. Atoms and their structures underpin all of chemistry. We'll also look at how electrons behave and the repercussions of their actions.

1.1 Discovery of fundamental particles

Dalton's atomic theory was successful in explaining the laws of conservation of mass, constant composition, and multiple proportion, but it failed to explain the results of numerous tests, such as the fact that solids like glass or ebonite create electricity when touched with silk or fur.

1.1.1 Electron discovery Of 1879, William Crookes investigated the electrical discharge in cathode ray discharge tubes, which are partially evacuated tubes. A discharge tube is a glass tube that is roughly 60cm long and has two thin metal bits called electrodes that are sealed inside. Crooke's tube is the name for this type of tube. The negative electrode is referred to as the cathode, whereas the positive electrode is referred to as the anode.

Invisible rays emanating from the cathode and producing a greenish light behind the perforated anode on the glass wall coated with phosphorescent substance ZnS are detected when a gas is confined at low pressure (10-4 atm) in a discharge tube is subjected to a high voltage (10,000V).

Cathode Ray Experiment

Cathode rays were the name given to these beams.

1.1.2 Characteristics

1. They cast a sharp shadow on solid objects in their route, implying that they are travelling in a straight line.

2. They are negatively charged because they are redirected towards the positive plate in an electric field. Stoney gave them the name electrons.

3. They can make a light paddle wheel that rotates in front of them. They have kinetic energy and are material particles, which means they have kinetic energy.

4. Their charge-to-mass ratio is 1.75882 1011C/kg.

5. They ionise the gases they pass through.

6. When they collide with a metallic target, they emit X-rays.

7. The material of the electrodes and the nature of the gas present in the cathode ray tube have no effect on the characteristics of cathode rays (electrons).

As a result, we can conclude that electrons are the fundamental building blocks of all matter.

1.1.3 Charge to mass ratio of electron

J.J. Thomson Experiment-Charge to Mass Ratio of Electron

J.J. Thomson used a cathode ray tube to measure the e/m ratio of electrons in 1897, employing electric and magnetic fields that were perpendicular to each other and to the path of electrons.

In the presence of an electric or magnetic field, the extent to which electrons deviate from their path is determined by: 

(a) the charge on the electron,

(b) the mass of the particle, and 

(c) the strength of the electric or magnetic field.

The electrons are deflected to point A when merely an electric field is applied. Electrons are deflected to point C when merely a magnetic field is provided. The electrons can hit the screen at point B by balancing the intensities of electric and magnetic fields.

1.1.4 Charge on the electron

To establish the charge on the electrons, R.A Millikan invented the oil drop experiment.

Millikan Invented the Oil Drop Experiment

Oil droplets in the form of mist were permitted to enter through a tiny hole in the upper plate of the electrical condenser in this manner. Through a microscope with a micrometre eye piece, the downward motion of these droplets was observed. Millikan was able to determine the mass of oil droplets by monitoring the velocity at which they fell. By passing an X-ray beam through the air inside the chamber, it was ionised. Collisions with gaseous ions gave these oil droplets their electrical charge.

Depending on the charge on the droplets and the polarity and strength of the voltage given to the plate, the fall of these charged oil droplets can be slowed, accelerated, or made immobile. Millikan discovered that the magnitude of electrical charge, q, on oil droplets is always an integral multiple of the electrical charge, e, by carefully monitoring the effects of electrical field intensity on the motion of oil droplets, q = n e, where n = 1, 2, 3…

The electron's charge is measured to be –1.6022 10–19C.

As a result, the mass of an electron was computed as

$$\text{m = }{\displaystyle \frac{\text{e}}{\text{e/m}}}={\displaystyle \frac{1.6022\times {10}^{-19}\text{C}}{1.758820{10}^{11}\text{C/ kg}}}=9.1094{10}^{-31}\text{ kg}$$

1.1.5 Where do cathode rays come from?

Because the gas molecules are bombarded by the high-speed electrons discharged initially from the cathode, cathode rays are produced first from the cathode material and subsequently from the gas inside the discharge tube.

1.2.1 Proton discovery

Because the atom is electrically neutral as a whole and the presence of negatively charged particles has been proved, it was assumed that some positively charged particles must also be present. As a result, during his research with cathode rays, physicist Goldstein devised a unique discharge tube. Canal rays are a new type of ray that he found.

Canal rays get their name from the fact that they move in the opposite direction of cathode rays in a vacuum tube, passing through and emerging from a canal or hole in the cathode.

Anode rays are another name for them.

1.2.2 Properties:

They travel in straight lines for the most part.

2. They have a positive charge on them.

3. They are formed up of particles of substance.

4. The charge on the particles that make up anode rays is discovered to be dependent on the type of gas used.

5. It is discovered that the mass of the particles that make up anode rays is affected by the type of gas used.

6. The particle's charge to mass ratio (e/m) is discovered to be dependent on the gas used.

7. Their behaviour in electric and magnetic fields is diametrically opposed to that of electrons.

1.2.3 Origin of anode rays: Anode rays are thought to be created when high-speed electrons from cathode rays strike gaseous atoms, knocking away their electrons. These anode rays are formed in the space between the anode and the cathode and are not released by the anode.

When hydrogen was used as the discharge gas, the lightest charged particles were obtained. These particles have the highest e/m value. They had the smallest mass and a single positive charge. A proton was the name given to the particle.

A proton's charge is $$+1.6022\times {10}^{-19}\text{C}$$

Mass of a proton = $$\text{1}\text{.672}\times {10}^{-27}\text{kg}$$

1.3.1 Neutron discovery

Rutherford proposed the theoretical requirement for the existence of a neutron particle in the atomic nucleus in 1920.

It was postulated that it be a neutral particle with a mass nearly equivalent to that of a proton. It was given the name neutron by him. Chadwick established its existence in 1932. When a beam of particles 2 4 (He) collided with Beryllium (Be), he saw that a new sort of particle was ejected. It was virtually the same mass as a proton (1.674 10–27kg) and had no charge.

$$\begin{array}{c}{}_5^{11}\text{B + }{}_2^4\text{He}\to {}_7^{14}\text{N + }{}_0^1\text{n}\\ {}_4^9\text{Be + }{}_2^4\text{He}\to {}_6^{12}\text{C + }{}_0^1\text{n}\end{array}$$

ATOMIC BUILDING BLOCKS

2. DEFINITIONS

2.1 Electron

A basic particle with a mass almost equivalent to 1/1837th of that of a hydrogen atom and one unit negative charge.

2.2 Proton

A basic particle with a mass almost equal to that of a hydrogen atom and one unit positive charge.

2.3 Neutron

A fundamental particle with a mass almost equal to that of a hydrogen atom but no charge.

3. MODEL THOMSON

Sir J. J. Thomson, the inventor of the electron, was the first to propose an atomic structure model.

(I) Electrons are found in every atom.

(ii) The entire atom is neutral. The total positive and negative charges must be the same.

He saw the atom's positive charge being evenly distributed around an atomic-sized sphere (about 10–10 m in diameter). The electrons were tiny particles with a negative charge equal to the positive charge in the atom when they were grouped together. They were like plums in a dessert, embedded in the atom. The charge distribution was such that the arrangement was the most stable. The plum - pudding model of the atom was popular at the time. Also available are the raisin pudding and watermelon models.

Plum - Pudding Model of the Atom

3.1 Drawbacks

Though the model was able to explain the atom's overall neutrality, it was unable to explain Rutherford's scattering experiments satisfactorily.

4. RUTHERFORD’S α-SCATTERING EXPERIMENT

In 1909, Rutherford conducted particle scattering experiments. A fine stream of alpha particles bombards an extremely thin gold foil (0.004nm) in this experiment. Behind the gold foil is a fluorescent screen (ZnS), on which points emerging from-particles were captured.

Polonium was employed as the -particles' source.

Rutherford’s α-Scattering Experiment

4.1 Observations

Rutherford conducted a series of tests utilising particle scattering by an extremely thin gold foil.

The following observations were made: 

(i) The majority of the particles (99 percent) pass through it without deflection or deviation.

(ii) Particles were deflected at tiny angles in several cases.

(iii) Only a few particles were deflected by considerable angles, and one particle was deflected by 180o on rare occasions.

4.2 Conclusions

(i) Because most particles flowed through it without deflection, an atom must be exceedingly hollow and made up mostly of empty space.

(ii) Only a few particles were deflected significantly. This means that: (a) Electrons cannot deflect heavy and positively charged particles due to their negative charge and low mass.

(b) There must be an extremely hefty and positively charged body in the atom, such as the nucleus, that prevents positively charged particles from passing through.

(c) Because the number of particles that experience 180o deflection is so small, the volume of a positively charged body must be a very small proportion of the atom's overall volume. This positively charged substance must be at the nucleus, which is the centre of the atom.

Scattering of  α-Particles

The radius of an atom is found to be in the range of 10–10m, while the radius of the nucleus is in the range of 10–15m. If a cricket ball represents a nucleus, the radius of an atom is around 5 kilometres5.3.1 Isotopes

.

4.3 Rutherford’s nuclear atomic model

Rutherford’s Nuclear Atomic Model

(i) At the centre of an atom lies a tiny positively charged nucleus, which is surrounded by a hollow region known as the extra nuclear part.

(ii) Nucleons, which are made up of protons and neutrons, give the nucleus its positive charge, whereas electrons, which are found in the extra nuclear region, have minimal mass, and carry a negative charge.

(iii) Because the number of electrons in the atom equals the number of protons, it is electrically neutral. As a result, the nucleus' total positive charge is balanced by the electrons' total negative charge.

(iv) In the extranuclear section, electrons orbit the nucleus in circular routes known as orbits. As a result, an atom mimics the solar system, with the sun acting as the nucleus and the planets acting as circling electrons, and the model is referred to as the planetary model.

(v) The electrostatic force of attraction holds electrons and nuclei together.

(vi) Centrifugal forces perfectly balance the forces of attraction acting on the electron.

4.4 Drawbacks

Drawbacks of Rutherford’s Nuclear Atomic Model

(i)Any charged entity in motion under the influence of attractive forces should radiate energy continually, according to classical mechanics. If this is the case, the electron will spiral and eventually fall into the nucleus, causing the structure to collapse. This type of behaviour is never seen.

(ii) It contains no information on the electronic structure of atoms, such as how electrons are dispersed around the nucleus and their energies.

5. ATOMIC NUMBER AND MASS NUMBER

5.1 Atomic number (Z)

The number of unit positive charges or protons contained in the nucleus of an element's atom is equal to the element's atomic number. It also represents the neutral atom's number of electrons. For example, the number of protons in Na is 11, hence the atomic number of Na is 11.

5.2 Mass number (A)

The nucleons are the elementary particles (protons and neutrons) that make up the nucleus of an atom.

An atom's mass number (A) is equal to the total of its protons and neutrons. It's always a number that's a whole number. As a result, the mass number (A) equals the sum of the number of protons (Z) and the number of neutrons (N)

Therefore, number of neutrons (n) = Mass Number (A) – Number of protons (Z) n = A – Z

The general notation that is used to represent the mass number and atomic number of a given atoms is ${}_Z^{A}X$

Where, $\text{X}-$ symbol of element

A - Mass number

$Z-$ atomic number

5.3 Isotopes, Isobars, isotones and Isoelectronic

5.3.1 Isotopes: Isotopes are the same atomic number but different mass number atoms of the same element. The discrepancy is due to the number of neutrons present.

The number of electrons in an atom determines its chemical characteristics. As a result, isotopes of the same element have the same chemical behaviour.

Isotopes of Hydrogen

Isotopes of Oxygen

Isotopes of Some Common Elements

5.3.2 Relative Abundance: In nature, varying percentages of an element's isotopes occur, which is referred to as relative abundance.

The average atomic mass of the element can be estimated using this relative abundance. Cl, for example, has an average atomic mass of 35.5 due to the presence of two isotopes, 35Cl and 37Cl, which are found in 75 percent and 25% abundance, respectively.

5.3.3 Isobars: Isobars are atoms of various elements with distinct atomic numbers but the same mass numbers.

5.3.4 Isotones: Isotones are atoms of various elements that have the same number of neutrons.

5.3.5 Isoelectronic: This term refers to species (atoms or ions) that have the same number of electrons. For example, $${{\text{O}}^{\text{2}}}^{\text{--}}\text{, }{\text{F}}^{\text{--}}\text{ , N}{\text{a}}^{\text{ + }}\text{ , M}{\text{g}}^{\text{ + 2}}\text{, A}{\text{l}}^{\text{ + 3}}\text{, Ne}$$, and so on.

To go deeper into the atomic secrets, we'll need to learn about electromagnetic radiations and Maxwell's Electromagnetic Wave Theory."

The first thorough description of the interaction between charged things and the behaviour of electric and magnetic fields was given by James Maxwell.

To go deeper into the atomic secrets, we'll need to learn about electromagnetic radiations and Maxwell's Electromagnetic Wave Theory."

The first thorough description of the interaction between charged things and the behaviour of electric and magnetic fields was given by James Maxwell.

6. ELECTROMAGNETIC RADIATIONS

Waves created by oscillating magnetic and electric fields that are perpendicular to each other and both perpendicular to the direction of motion are known as electromagnetic radiation.

Electromagnetic Wave Graph

They do not require any medium and, unlike sound waves, can move in a vacuum.

Light is a type of radiation with wave properties.

The following are some of the characteristics of a wave:

1) Amplitude: The height of a wave's crest or trough (depth). metre (metric) (m)

2) Frequency: The number of waves that travel through a certain place in a second. Hertz (Hz) or s–1 are the units of measurement.

3) Time Period: A wave's period is the amount of time it takes to complete one vibration. time in seconds

4) Velocity: The velocity of a wave is the distance it travels in one second. m/s is the unit of speed.

All types of electromagnetic radiation travel at the same speed in vacuum, which is 3 108 m/s. This is known as the speed of light.

5) Wavelength(: A wavelength is the distance between two consecutive crests or troughs. Angstrom() [1=10–10m] is a unit of measurement.

6) Wave Number: This is the number of wavelengths per centimetre. m-1 is the unit of measurement.

$$\bar{\nu }=1/\lambda$$

6.1 Relationship between Velocity, Frequency & Wavelength

c = $\nu .\lambda$

$$\text{c}$$: speed of light i.e. 3 × 108 m/s in vaccum 

$$\nu$$: frequency;

 λ : wavelength

7. ELECTROMAGNETIC SPECTRUM

The electromagnetic spectrum is the band of radiations created when all electromagnetic radiations are grouped in ascending order of wavelength or decreasing frequency.

Electromagnetic Spectrum 

The visible spectrum (VIBGYOR) is a subset of this spectrum with a wavelength range of 380-760 nm.

In this order, the wavelengths increase:

Gamma Rays < X-rays < Ultra-violet rays < Visible< Infrared < Micro-waves

7.1 Electromagnetic Wave Theory

(1) A source (such as a heated rod) emits energy continuously in the form of radiations (i.e., there is no change in wavelength or frequency of the emitted radiations as the energy radiated increases).

(2) These are electromagnetically emitted radiations.

7.2 Failure of EM wave theory

Two experiments shattered the theory:

(1) Black Body Radiation: According to Maxwell's theory of body heating, the intensity, or the amount of energy radiated per unit area, should grow without affecting the wavelength or frequency.

However, we notice that when we heat an iron rod, it turns red, then white, and finally blue at extremely high temperatures. The frequency of radiated radiations is shifting because of this.

A black body is an ideal body that produces and absorbs radiation of all frequencies, and the radiation emitted by a black body is known as black body radiation.

The intensity of a black solid varies with wavelength at different temperatures, as seen below:

Variation of Intensity of a Black Solid Vs Wavelength at Different Temperatures

As a result, as the temperature rises, the predominate wavelength in the emitted radiations drops while the frequency rises.

At greater temperatures, the intensity increases as predicted by Maxwell's theory, but the wavelength shortens. If $${\text{T}}_1>{\text{T}}_2>{\text{T}}_3$$, the answer is $${\lambda }_1<{\lambda }_2<{\lambda }_3$$.

Photoelectric Effect No. 2

When radiations with a specific minimum frequency (0) strike a metal's surface, electrons are expelled from the metal's surface. The photoelectric effect is the name for this phenomenon. Photoelectrons are the electrons that are released.

According to Maxwell's Theory, electron ejection should be dependent on the intensity of radiation; if electrons are not ejected, raising the intensity will allow them to be ejected.

Photoelectric Effect

Following are some observations:

(i) The electrons are ejected from the metal surface as soon as the light beam impacts it, i.e. there is no time lag between the light beam impacting the metal surface and the ejection of electrons.

(ii) The intensity or brightness of light is related to the number of electrons ejected.

(iii) There is a distinctive minimum frequency, 0 (also known as threshold frequency), below which no photoelectric effect is detected for each metal. The expelled electrons have a particular kinetic energy when the frequency is greater than zero. The kinetic energy of these electrons grows as the frequency of light utilised increases.

As a result, these observations ran counter to Maxwell's theory. The number of electrons ejected, and the accompanying kinetic energy should be proportional to the intensity of light. Although the number of electrons emitted is proportional to the brightness of the light, the kinetic energy of the ejected electrons is not.

Max Von Planc's Quantum theory was developed to support these discoveries.

8. PLANCK’S QUANTUM THEORY

The essential points of this theory are:

(i) Energy is emitted or absorbed discontinuously in the form of small discrete packets of energy rather than constantly. A 'quantum' is the name given to each such energy packet.

This quantum of energy is known as a photon in the case of light.

Planck’s Quantum Theory: Energy is Emitted or Absorbed Discontinuously in the Form of Small Discrete Packets of Energy Rather than Constantly

(ii) Only one photon can be delivered to one electron or other particle at a time.

(iii) It is impossible to divide or distribute a quantum.

(iv) The frequency of radiation is precisely proportional to the energy of each quantum.

$$\text{E}\alpha \nu \text{ or E = hv = }{\displaystyle \frac{\text{hc}}{\lambda }}$$

h = Planck’s constant = $$6.626\times {10}^{-34}\text{Js}$$

(v) A body's total energy emitted or absorbed will be quantified in whole numbers.

$$\text{E = nhv = }{\displaystyle \frac{\text{nhc}}{\lambda }}$$

This is also known as "energy quantisation."

8.1 Explanation of Black Body Radiation

The energy emitted increases as the temperature rises, increasing the frequency of the emitted radiations.

The wavelength moves to lower values as the frequency rises.

8.2 Explanation of Photoelectric effect

(i) When a photon of a specific frequency strikes the surface of a metal atom, it imparts all its energy to the metal atom's electron. Only if the photon's energy is high enough to overcome the nucleus's attraction to the electron will the electron be ejected from the metal. As a result, photoelectrons are only expelled when the input light has a minimal frequency (threshold frequency 0). The "Work Function," which is "$${\text{h}}_0$$," is the threshold energy necessary for emission.

(ii) If the incident light's frequency ($${\text{v}}_0$$) exceeds the threshold frequency ($${\text{v}}_0$$), the surplus energy is transferred to the electron as kinetic energy.

Energy of one quantum = Threshold Energy + Kinetic Energy

$$hv=h{v}_0\text{ + }\left(1/2\right)m_c{v}^2$$

(iii) When, the number of quanta incident increases as the intensity increases, resulting in an increase in the number of photoelectrons expelled.

(iv) When $$v_0>v$$, the energy of each photon increases, and consequently the kinetic energy of each ejected electron grows as the frequency is increased.

Note:

Energy can also be expressed in Electron Volt(eV).

The energy acquired by an electron when it is accelerated through a potential difference of one volt.

$1\text{eV}=1.602\times {10}^{-19}\text{J}$

CONCLUSION:

(iii) When, the number of quanta incident increases as the intensity increases, resulting in an increase in the number of photoelectrons expelled.

(iv) When $$v_0>v$$, the energy of each photon increases, and consequently the kinetic energy of each ejected electron grows as the frequency is increased.

9. WHAT IS SPECTRUM?

A spectrum is a group or band of wavelengths/colors, and spectroscopy is the study of emission or absorption spectra.

9.1 Types of spectrum

Types of spectrums are

1) Emission Spectrum 

2) Absorption Spectrum

9.2 Emission Spectrum

When rays from a source strike a prism and are split into distinct wavelengths before being recorded on a photographic plate.

(a) Continuous Emission Spectra: There are no pauses between wavelengths, and one merges into the next.

Continuous Emission Spectra of White Light

Discontinuous Emission Spectra, also known as Line Spectra or Atomic Spectra, are a type of continuous emission spectrum.

Certain wavelengths are missing from a group, leaving dark areas in between, causing the spectrum to be discontinuous. It's also known as an element's fingerprint.

Line Spectrum Produced Form a Volatile Salt Placed in a Flame

9.3 Absorption Spectra

When light from any source is first passed through the solution of a chemical substance and then analysed, it is observed that there are some dark lines in the otherwise continuous spectra.

Absorption spectrum

Bohr researched the atomic spectra of hydrogen and offered his hypothesis based on his findings.

10. BOHR’S MODEL

10.1 Postulates

1) An atom is made up of a small, heavy, positively charged nucleus at the centre, which is surrounded by electrons in circular orbits.

2) Electrons can only revolve in orbits with a predetermined energy value. As a result, these orbits are referred to as "energy levels" or "stationary states."

1,2,3,... are the numbers assigned to them. Principal Quantum Numbers are the names given to these numbers.

(a) An electron's energy is given by:

$${\text{E}}_{\text{n}}=\text{--}{\text{R}}_{\text{H}}\left({\text{Z}}^{\text{2}}\text{/}{\text{n}}^{\text{2}}\right)\text{n = 1,2,3}.......$$

where $${\text{R}}_{\text{H}}$$ is Rydberg’s constant and its value is 2.18 × 10–18 J.

Z = atomic number

${\text{E}}_{\text{n}}=-2.18\times {10}^{-18}{\displaystyle \frac{{\text{Z}}^2}{{\text{n}}^2}}\text{J/atom}$

  ${\text{E}}_{\text{n}}=-13.6{\displaystyle \frac{{\text{Z}}^2}{{\text{n}}^2}}\text{eV/atom}$

 ${\text{E}}_{\text{n}}=-1312{\displaystyle \frac{{\text{Z}}^2}{{\text{n}}^2}}\text{kJ/mol}$

As a result, the energies of various levels are arranged in the following order: K L M N...... and so on.

The energy of the lowest state (n=1) is referred to as ground state energy.

(b) Stationary state radiuses:

$${\text{r}}_{\text{n}}=-{\displaystyle \frac{\text{52}\text{.9}{\text{n}}^2}{{\text{Z}}^2}}\text{pm}$$

The radius of the first stationary state for an H-atom (Z = 1) is known as the Bohr orbit (52.9 pm)

(c) Electron velocities in different orbits: 

$${\text{v}}_{\text{n}}={\displaystyle \frac{2.188\times {10}^2}{\text{n}}}\text{m/s}$$

3) Because electrons can only rotate in orbits with fixed energy values, electrons in an atom can only have certain certain energy values and not any of their own. An electron's energy is so quantified.

4) An electron in an atom's angular momentum, like its energy, can take on specific definite values and not any value at all.

$$\text{mvr = }{\displaystyle \frac{\text{nh}}{2\pi }}$$

Where n=1,2,3...... 

5) When an electron is present in the same shell, it does not lose or gain energy.

6) When an electron obtains energy, it is excited to higher energy levels, and when it loses energy in the form of electromagnetic radiation, it is de-excited and returns to lower energy levels.

Excitation by Absorption of Light and De-Excitation by Emission of Light

10.2 What does negative energy in the case of the hydrogen atom imply?

The energy of an electron in an atom is lower than the energy of a free electron at rest, as shown by the negative sign. The energy value of a free electron at rest is zero since it is infinitely far away from the nucleus ($$\text{n = }\mathrm{\infty }$$). En becomes increasingly negative as the electron approaches closer to the nucleus (as n decreases). n=1 corresponds to the most stable orbit and has the largest negative energy value.

Note: Bohr’s model is applicable to H-atom or H-like species like $$\text{H}{\text{e}}^{+}\text{,L}{\text{i}}^{2+}\text{,B}{\text{e}}^{3+}$$

10.3 Transition of Electron

$\mathrm{\Delta }E={\text{E}}_{\text{f}}\text{ - }{\text{E}}_{\text{i}}$

$\mathrm{\Delta }E=\left[-{\displaystyle \frac{{\text{R}}_{\text{H}}}{{\text{n}}_{\text{f}}^2}}\right]-\left[-{\displaystyle \frac{{\text{R}}_{\text{H}}}{{\text{n}}_{\text{i}}^2}}\right]$

$\mathrm{\Delta }E={\text{R}}_{\text{H}}\left[{\displaystyle \frac{1}{{\text{n}}_{\text{i}}^2}}-{\displaystyle \frac{1}{{\text{n}}_{\text{f}}^2}}\right]=2.18\times {10}^{-18}\left[{\displaystyle \frac{1}{{\text{n}}_{\text{i}}^2}}-{\displaystyle \frac{1}{{\text{n}}_{\text{f}}^2}}\right]\text{J/atom}$

The wavelength associated with the photon's absorption or emission is:

$${\displaystyle \frac{\text{1}}{\lambda }}\text{ = }{\displaystyle \frac{\mathrm{\Delta }E}{\text{hc}}}\text{ = }{\displaystyle \frac{{\text{R}}_{\text{H}}}{\text{hc}}}\left[{\displaystyle \frac{1}{{\text{n}}_{\text{i}}^2}}-{\displaystyle \frac{1}{{\text{n}}_{\text{f}}^2}}\right]=1.09677\times {10}^7\left[{\displaystyle \frac{1}{{\text{n}}_{\text{i}}^2}}-{\displaystyle \frac{1}{{\text{n}}_{\text{f}}^2}}\right]{\text{m}}^{-1}$$