Summarize result (100%)

The Nuclear Model

At the end of the 1800s, J.J. Thomson's discovery of the electron led to the understanding of atoms as containing subatomic particles.- Example: This model is often likened to a "plum pudding" model.

Subtopic 2: Rutherford's Experiment

Key Concept: Rutherford's experiment involved bombarding a thin gold foil with alpha particles to investigate the distribution of mass and charge in the atom. - Example: Most of the atom's mass was found to be concentrated in a small volume at the atom's center, leading to the proposal of a nuclear model of the atom.

Subtopic 3: Alpha Particles

Key Concept: Alpha particles were found to be massive, positively charged particles emitted during radioactive decay. - Example: These particles were detected by the small flashes of light when they collided with a zinc-sulfide-coated screen. ## Subtopic 4: Rutherford's Conclusion
Key Concept: Rutherford's experiment led to the conclusion that the atom had a small, dense, positively charged nucleus # Rutherford's Gold Foil Experiment

Surprising Results

Rutherford expected minor deflections of alpha particles passing through gold foil. - Example: Like expecting a light breeze to move a beach ball. ## Unexpected Observations

Major Deflections: Some particles were scattered at large angles, even >90?. 2. Rebounds: A few particles rebounded at large angles. - Reflection: Rutherford likened this to firing a cannon at tissue paper and having it bounce back. ## Revised Model

Nuclear Model: Rutherford proposed a tiny, massive nucleus with concentrated positive charge. - Example: Similar to a cannonball hitting tissue paper and bouncing back. - Insight: This led to the understanding that most of an atom's mass and positive charge is in the nucleus. ## Emission Spectra: Understanding Atomic Arrangement
Key Concept: Emission spectra are unique patterns of light emitted by atoms, used to identify elements. - Example: Gas-discharge tubes demonstrate this with different gases emitting unique colors. ## Gas-Discharge Tubes and Emission Spectra

Principle: Gas-discharge tubes contain low-pressure gas and electrodes, emitting light when a potential difference is applied. 2. Unique Colors: Different gases emit unique colors, forming a characteristic emission spectrum. - Example: Passing emitted light through a diffraction grating or prism produces an emission spectrum, unique to each element. - The spectrum is a series of distinct lines of different colors. Each line corresponds to a particular wavelength of light emitted by the atoms of that gas

Spectroscopy and Emission Spectrum Analysis

Instrument: A spectroscope is used to study emission spectra in greater detail. - Process: Light passes through a slit, diffracts through a grating, and forms an image at different positions for each wavelength. # NOTE :
Emission Spectrum:
- This term refers to the specific pattern of light (wavelengths) emitted by atoms or molecules of an element when they are excited. Each element has its unique emission spectrum, which acts like a "fingerprint."Energy Levels of the Hydrogen Atom:
- Substituting the expression for rnr_nrn into the equation for energy, we get :
  
  En=-Ke^4me/2h^2 x 1/n^2
- Substituting numerical values:
  
  En=-2.17x10-18 Jx1/n2
- Converting the energy from joules to electron volts (1 eV=1.6x10-19 J):
En=-13.6eVxn21

Summary

Bohr Radius (n=1n ): r1=0.053 nm
Energy Levels:

En=-13.6 eVx1n2
- The energy is quantized and is inversely proportional to the square of the principal quantum number n.
- The lowest energy state (n=1) has an energy of -13.6eV, and as n increases, the energy levels get closer together.- When an electron transitions between these energy levels, it either absorbs or emits a photon with an energy equal to the difference between the initial and final states: ?E=Efinal-Einitial

Visible, Ultraviolet, and Infrared Spectral Lines:

The hydrogen atom can emit electromagnetic radiation in different regions of the spectrum:
- Ultraviolet light is emitted when the electron falls to the ground state (n=1) from a higher energy level.- When an atom absorbs a photon, it gains energy equal to that of the photon, and when it returns to a lower energy state, it emits a photon with a characteristic energy

Energy Change in Atomic Transitions:

When an atom transitions from an initial energy level Ei to a final energy level Ef, the change in energy (?Eatom) is calculated by: ?Eatom=Ef-Ei
If the energy decreases (a transition to a lower energy level), the energy difference corresponds to the energy of the emitted photon.Deriving the Orbital Radius:
- Using Bohr's quantization of angular momentum and the above relationship, we can solve for the orbital radius
  
  rn=n2h24?2Kmee2
- Substituting known values for constants and n=1n = 1n=1 (for the ground state)
  
  r1=(6.626x10-34 J?s)2x124?2(9.0x109 N?m2/C2)(9.11x10-31 kg)(1.60x10-19 C)2r_1 = \frac{(6.626 \times 10^{-34} , \text{J} \cdot \text{s})^2 \times 1^2}{4 \pi^2 (9.0 \times 10^9 , \text{N} \cdot \text{m}^2/\text{C}^2) (9.11 \times 10^{-31} , \text{kg}) (1.60 \times 10^{-19} , \text{C})^2}
  
  r1=5.3x10-11 m or 0.053 nmr
- This is known as the Bohr radius.### Final Summary:
Key Takeaways: Emission spectra show emitted light wavelengths, aiding in identification, while absorption spectra reveal absorbed light, enabling analysis of element composition.- Relationship for Orbital Radius:
- By combining the quantization of angular momentum and Coulomb's force, Bohr derived an expression for the radius of the electron's orbit for different energy levels in a hydrogen atom.Photon Energy Equation:
The energy of the emitted photon Ephoton is equal to the negative of the change in the atom's energy: Ephoton=-?Eatom
Alternatively, it can be expressed as: Ephoton=Ei-Ef

Planck's Equation for Photon Energy:
- The energy of the emitted photon is also given by Planck's equation: Ephoton=hf
- Here, h is Planck's constant, and f is the frequency of the emitted photon.# NOTE
Summary Insight: Both emission and absorption spectra are crucial tools for identifying and analyzing elements and compounds in various fields.- Application to Hydrogen Spectrum:
- Bohr's model accurately predicted the wavelengths of light emitted by a hydrogen atom, matching experimental measurements by other scientists.## Determining Star Composition
Key Concept: Scientists determine the composition of stars by comparing the missing lines in the observed spectrum with the known emission spectra of various elements.- Aimed to resolve the issues in the nuclear model by incorporating Planck's concept of quantized energy levels and Einstein's theory of light

Stable Condition and Stationary States

Bohr proposed that the laws of electromagnetism do not apply inside the atom.Quantization of Angular Momentum:
- According to Bohr's model, the angular momentum of an electron orbiting the nucleus is quantized and is given by the equation
  
  mevr=nh/2?## Subtopic 1: Thomson's Model
Key Concept: J.J. Thomson proposed that the atom contained a massive, positively charged substance with negatively charged electrons distributed throughout, similar to raisins in a muffin.# Absorption Spectra

Key Concept:

Absorption Spectra are produced when white light passes through a gas sample and then through a prism, resulting in a continuous spectrum with dark lines corresponding to absorbed wavelengths.Analyzing Unknown Materials: Spectroscopy allows scientists to analyze and identify unknown materials by observing their emission and absorption spectra.Industrial Applications: Industries like steel mills use spectroscopy to determine the composition of materials, such as scrap iron, for commercial purposes.Coulomb's Law (Force between Electron and Nucleus):
- The centripetal force keeping the electron in orbit is provided by the electrostatic attraction between the positively charged nucleus and the negatively charged electron:
- F = Ke^2/r^2
  
  where:
  - K = Coulomb's constant (9.0x109N?m2/C2)
  - e = charge of the electron (1.60x10-19C) 3.### The Significance of the Bohr Model:
Even though the Bohr model could not accurately predict the spectral lines of atoms with more than one electron, it was crucial in advancing the understanding of atomic structure.- Contributions of the Bohr Model:
- It accurately predicted the emission spectrum of hydrogen, including the precise wavelengths of its spectral lines.- The emission spectrum shows which wavelengths of light an atom or molecule emits when its electrons transition from higher to lower energy levels.## Emission Spectra
Spectrum of an incandescent solid is a continuous band of colors from red through violet
Key Concept: Emission spectra show the wavelengths of light emitted by atoms of a gas.- Example: Mercury and neon emission spectra display distinct colored lines corresponding to specific wavelengths.- When the emission spectrum of a combination of elements is photographed, an analysis of the lines on the photograph can indicate the identities and the relative concentrations of the elements present.- Using Einstein's photoelectric theory, Bohr determined that the energy of each emitted photon is EPhoton=hf, where h is Planck's constant and f is the frequency of the photon.- Limitations of the Bohr Model:
- The model only worked for hydrogen and couldn't predict the spectrum of helium, the next simplest element.where:
  - me = mass of the electron (9.11x10-31kg)
    
    9.11x10-31 kg9.11 \times 10^{-31} , \text{kg}
  - v = velocity of the electron
  - r = radius of the orbit
  - n = principal quantum number (integer values: 1, 2, 3, ...)
  - h = Planck's constant (6.626x10-34J?s)

2.Calculating the Total Energy of the Electron: - The total energy of the electron in orbit is the sum of its kinetic energy (KE) and potential energy (PE)

    E=KE+PE
    
- The kinetic energy is given by

KE= 1/2mv^2

- The potential energy due to electrostatic attraction is:

PE=-Ke^2/r

- Using KE=-1/2PE the total energy E is :

E=-Ke^2/2r

6.- It provided a method to calculate the ionization energy of a hydrogen atom, which matched experimental data closely.- Bohr's insights into electron arrangements also influenced our understanding of chemical bonding and the periodic table, emphasizing that each element has a unique electron configuration.### Historical Impact:

Niels Bohr received the Nobel Prize in Physics in 1922 for his work on the structure of atoms and their radiation, marking a major step in the development of modern atomic theory.## Observation of Fraunhofer Lines
Key Concept: Fraunhofer lines are dark lines observed in the spectrum of sunlight, caused by gases in the Sun's atmosphere absorbing specific wavelengths of light.- Example: When sunlight passes through the Sun's atmosphere, certain gases absorb characteristic wavelengths, creating dark lines in the spectrum.- Moving an electron farther from the nucleus requires work, so atoms with electrons in distant orbits have higher energy.ELECTRON ORBITAL IN HYDROGEN :

rn= h^2(n^2)/4pi^2Kme(e^2)

Bohr's Model of the Hydrogen Atom

1.rn?n2r_n \propto n^2

- **Energy** decreases in magnitude as En?-1/n2
    
    En?-1/n2E_n \propto -1/n^2

Why is the Energy Negative in the Bohr Model?### Energy Levels and Spectral Lines:

The energy levels in a hydrogen atom are quantized and given by the formula: En=-13.6eVx1/n^2 where n is the principal quantum number.- Had a major flaw: according to electromagnetic theory, an accelerating electron (moving in a curved path) should lose energy by emitting radiation.- Bohr postulated that the energy levels in an atom are quantized, meaning they exist in specific, discrete amounts.## Quantization of Energies
The quantization of energies of electrons in atoms can be likened to a flight of stairs with decreasing-height steps.- This concept forms the basis of Bohr's atomic model, where electrons orbit the nucleus in fixed, quantized energy levels.- Bohr's Explanation of Atomic Emission Spectra:
- Bohr proposed that stable atoms do not radiate energy while in a fixed orbit.### Predictions of the Bohr Model
Predictive Power of a Theory:
- Bohr's model provided predictions that could be tested experimentally, a key aspect of a strong scientific theory.- Coulomb's law for the force F between an electron (charge -e) and a proton (charge +e) separated by distance r is given by: F=-Ke^2/r^2
- The centripetal acceleration of the electron is a=r/v^2, where v is the electron's velocity.- Quantization of Angular Momentum:
- Bohr proposed that the electron's angular momentum is quantized, meaning it can only have certain discrete values.- The four visible lines in the hydrogen spectrum (red, blue-green, blue-violet, and violet) correspond to transitions from n=3,4,5,and6->n=2, respectively.For instance, astronomers use the emission spectra of distant stars to determine their chemical composition.- The practical application of emission spectra lies in their uniqueness, making it possible to differentiate and identify elements based on their emitted light.Corresponding Bright and Dark Lines: For a gas, the bright lines of the emission spectrum and the dark lines of the absorption spectrum occur at the same wavelengths.## Revising Rutherford's Nuclear model
19th Century Studies of Atomic Spectra:
- Physicists focused on atomic spectra to understand atomic structure.- Hydrogen, the simplest element with a simple spectrum, was studied extensively.- This energy loss would cause the electron to spiral into the nucleus within a fraction of a second, making atoms unstable.- Problems with the Planetary Model:
- Contradicted the stability of atoms, as it implied atoms would collapse.- However, atoms emit light only at specific wavelengths (seen in atomic spectra), which the planetary model could not explain.- Quantization of Energy and Orbits:
- Because energy is quantized, the electron orbits (distances from the nucleus) are also quantized.- Therefore, the energy of an emitted photon is both the product of Planck's constant and frequency, as well as the decrease in the atom's energy.- Here, me is the electron's mass, r is the radius of the orbit, h is Planck's constant, and n is an integer (the principal quantum number).- For a hydrogen atom, Bohr's model calculates the ionization energy as the energy difference between the ground state (n=1) and the ionized state (n->?).- The model introduced the concept of quantized electron orbits, which laid the groundwork for quantum mechanics.- Use of Emission Spectra:
- Emission spectra are used as a tool to identify elements, because each element's spectrum is unique.These spectra are used to identify unknown samples or gases by comparing their emitted wavelengths with those of known gases.# Unknown Samples
An unknown gas can be identified by comparing its wavelengths with the wavelengths measured in the spectra of known gas samples.- The elements that are present in the greatest concentrations produce emission lines of greater relative intensities.Gas Identification: The set of wavelengths absorbed by a gas forms its absorption spectrum.- Example: Gaseous elements absorb the same wavelengths that they emit when excited.Composition Determination: The composition of a gas can be determined from the wavelengths of the dark lines in its absorption spectrum.- Example: By analyzing the dark lines in the spectrum, scientists can identify the gases present in the star's atmosphere.- Rutherford's Nuclear Model:
- Proposed that electrons orbit the nucleus like planets orbit the Sun.- Predicted that electrons would emit energy continuously at all wavelengths.- An electron in a stable orbit in an atom does not radiate energy, despite accelerating.## Energy of an Atom:
The total energy of an atom is the sum of the kinetic energy of the electrons and the potential energy due to the attraction between electrons and the nucleus.- This agreement between prediction and experiment led to widespread acceptance of Bohr's model for hydrogen.### Development of Bohr's Model
Application of Newton's Second Law and Coulomb's Law:
- Bohr used Newton's second law of motion (F=ma) and Coulomb's law to describe the electron's motion around the nucleus.Relationship Between Centripetal Force and Coulomb's Force:
- Equating the centripetal force to the Coulomb force:
  
  mev2r=Ke2r2

4.### Visual Representation

For n=1,2,3,...:
- Radius increases as rn?n2.This negative sign indicates that the electron is in a bound state, meaning it is attracted to the nucleus and requires energy to be freed from this attraction.### Understanding Zero Energy:
The zero energy level is defined as the point where the electron is infinitely far from the nucleus, having no kinetic energy (i.e., it is at rest).In this state, the atom is ionized, and there is no interaction between the nucleus and the electron.The energy is negative because it represents the energy required to pull the electron from its orbit (bound state) to a state of zero interaction (ionization).E1=-13.6eV Thus, to ionize a hydrogen atom from its ground state, 13.6 eV of energy is required.- Through a comparison of the line intensities, the percentage composition of the gaseous material can be determined.- The visible hydrogen spectrum has four main lines: red, green, blue, and violet.- Niels Bohr's Contribution:
- Joined Rutherford's group in England in 1911.- Electromagnetic energy is emitted only when an electron transitions from a higher energy level (excited state) to a lower energy level.- A successful theory should not only explain observed phenomena but also make accurate predictions.In the Bohr model, the total energy of an electron in orbit around a nucleus is negative.- Infrared light is emitted when the electron falls to the n=3 or higher levels (e.g., n=4,5,...).For example, the Sun was found to be mostly hydrogen and helium using this technique.- Electrons have quantized amounts of energy, each called an energy level.- It also lacked a strong explanation for why electromagnetic laws should work everywhere except within the atom itself.- The quantization condition is:
  
  mevr=nh/2?- Therefore, any electron bound to the nucleus (i.e., in any orbit) has a total energy less than zero.### Ionization Energy:
Ionization energy is the amount of energy needed to completely remove the electron from the atom, taking it from a bound state to a free state.- Visible light is emitted when the electron falls to the n=2 level from a higher level (n>=3).- Scientists had questions about the location and nature of the atom's mass, leading to the development of the nuclear model.## Importance of Spectroscopy

1.- Any theory on atomic structure needed to explain these specific wavelengths and support the nuclear model.- An electron's energy cannot have a value between allowed energy levels.- The smallest allowable amount of energy is the ground state, while any level above it is an excited state.- Electrons closer to the nucleus have lower energy, while electrons further away have higher energy (excited states).This set of transitions is known as the Lyman series.This set of transitions is known as the Balmer series.This set of transitions is known as the Paschen series (for n=3) and others.- This stable condition is called a stationary state, with each state having its own specific amount of energy.## Practice & Application

1./2c 2.3.2.5.

Original text

The Nuclear Model

At the end of the 1800s, J.J. Thomson's discovery of the electron led to the understanding of atoms as containing subatomic particles.

Scientists had questions about the location and nature of the atom's mass, leading to the development of the nuclear model.

Subtopic 1: Thomson's Model

Key Concept: J.J. Thomson proposed that the atom contained a massive, positively charged substance with negatively charged electrons distributed throughout, similar to raisins in a muffin.
- Example: This model is often likened to a "plum pudding" model.

Subtopic 2: Rutherford's Experiment

Key Concept: Rutherford's experiment involved bombarding a thin gold foil with alpha particles to investigate the distribution of mass and charge in the atom.
- Example: Most of the atom's mass was found to be concentrated in a small volume at the atom's center, leading to the proposal of a nuclear model of the atom.

Subtopic 3: Alpha Particles

Key Concept: Alpha particles were found to be massive, positively charged particles emitted during radioactive decay.
- Example: These particles were detected by the small flashes of light when they collided with a zinc-sulfide-coated screen.

Subtopic 4: Rutherford's Conclusion

Key Concept: Rutherford's experiment led to the conclusion that the atom had a small, dense, positively charged nucleus # Rutherford's Gold Foil Experiment

Surprising Results

Rutherford expected minor deflections of alpha particles passing through gold foil.
- Example: Like expecting a light breeze to move a beach ball.

Unexpected Observations

Major Deflections: Some particles were scattered at large angles, even >90°.

Rebounds: A few particles rebounded at large angles.
- Reflection: Rutherford likened this to firing a cannon at tissue paper and having it bounce back.

Revised Model

Nuclear Model: Rutherford proposed a tiny, massive nucleus with concentrated positive charge.
- Example: Similar to a cannonball hitting tissue paper and bouncing back.

Insight: This led to the understanding that most of an atom's mass and positive charge is in the nucleus.

Emission Spectra: Understanding Atomic Arrangement

Key Concept: Emission spectra are unique patterns of light emitted by atoms, used to identify elements.
- Example: Gas-discharge tubes demonstrate this with different gases emitting unique colors.

Gas-Discharge Tubes and Emission Spectra

Principle: Gas-discharge tubes contain low-pressure gas and electrodes, emitting light when a potential difference is applied.

Unique Colors: Different gases emit unique colors, forming a characteristic emission spectrum.
- Example: Passing emitted light through a diffraction grating or prism produces an emission spectrum, unique to each element.
- The spectrum is a series of distinct lines of different colors. Each line corresponds to a particular wavelength of light emitted by the atoms of that gas

Spectroscopy and Emission Spectrum Analysis

Instrument: A spectroscope is used to study emission spectra in greater detail.

Process: Light passes through a slit, diffracts through a grating, and forms an image at different positions for each wavelength.

NOTE :

Emission Spectrum:
- This term refers to the specific pattern of light (wavelengths) emitted by atoms or molecules of an element when they are excited. Each element has its unique emission spectrum, which acts like a "fingerprint."
- The emission spectrum shows which wavelengths of light an atom or molecule emits when its electrons transition from higher to lower energy levels.

Use of Emission Spectra:
- Emission spectra are used as a tool to identify elements, because each element’s spectrum is unique. For instance, astronomers use the emission spectra of distant stars to determine their chemical composition.
- The practical application of emission spectra lies in their uniqueness, making it possible to differentiate and identify elements based on their emitted light.

Emission Spectra

Spectrum of an incandescent solid is a continuous band of colors from red through violet

Key Concept: Emission spectra show the wavelengths of light emitted by atoms of a gas.
- Example: Mercury and neon emission spectra display distinct colored lines corresponding to specific wavelengths.

These spectra are used to identify unknown samples or gases by comparing their emitted wavelengths with those of known gases.

Unknown Samples

An unknown gas can be identified by comparing its wavelengths with the wavelengths measured in the spectra of known gas samples.

When the emission spectrum of a combination of elements is photographed, an analysis of the lines on the photograph can indicate the identities and the relative concentrations of the elements present.

The elements that are present in the greatest concentrations produce emission lines of greater relative intensities.

Through a comparison of the line intensities, the percentage composition of the gaseous material can be determined.

Absorption Spectra

Key Concept:

Absorption Spectra are produced when white light passes through a gas sample and then through a prism, resulting in a continuous spectrum with dark lines corresponding to absorbed wavelengths.

Practice & Application

Gas Identification: The set of wavelengths absorbed by a gas forms its absorption spectrum. /2c

Corresponding Bright and Dark Lines: For a gas, the bright lines of the emission spectrum and the dark lines of the absorption spectrum occur at the same wavelengths.
- Example: Gaseous elements absorb the same wavelengths that they emit when excited.

Composition Determination: The composition of a gas can be determined from the wavelengths of the dark lines in its absorption spectrum.

NOTE

Summary Insight: Both emission and absorption spectra are crucial tools for identifying and analyzing elements and compounds in various fields.

Final Summary:

Key Takeaways: Emission spectra show emitted light wavelengths, aiding in identification, while absorption spectra reveal absorbed light, enabling analysis of element composition.

Observation of Fraunhofer Lines

Key Concept: Fraunhofer lines are dark lines observed in the spectrum of sunlight, caused by gases in the Sun's atmosphere absorbing specific wavelengths of light.
- Example: When sunlight passes through the Sun's atmosphere, certain gases absorb characteristic wavelengths, creating dark lines in the spectrum.

Determining Star Composition

Key Concept: Scientists determine the composition of stars by comparing the missing lines in the observed spectrum with the known emission spectra of various elements.
- Example: By analyzing the dark lines in the spectrum, scientists can identify the gases present in the star's atmosphere. For example, the Sun was found to be mostly hydrogen and helium using this technique.

Importance of Spectroscopy

Analyzing Unknown Materials: Spectroscopy allows scientists to analyze and identify unknown materials by observing their emission and absorption spectra.

Industrial Applications: Industries like steel mills use spectroscopy to determine the composition of materials, such as scrap iron, for commercial purposes.

Revising Rutherford’s Nuclear model

19th Century Studies of Atomic Spectra:
- Physicists focused on atomic spectra to understand atomic structure.
- Hydrogen, the simplest element with a simple spectrum, was studied extensively.
- The visible hydrogen spectrum has four main lines: red, green, blue, and violet.
- Any theory on atomic structure needed to explain these specific wavelengths and support the nuclear model.

Rutherford’s Nuclear Model:
- Proposed that electrons orbit the nucleus like planets orbit the Sun.
- Had a major flaw: according to electromagnetic theory, an accelerating electron (moving in a curved path) should lose energy by emitting radiation.
- This energy loss would cause the electron to spiral into the nucleus within a fraction of a second, making atoms unstable.

Problems with the Planetary Model:
- Contradicted the stability of atoms, as it implied atoms would collapse.
- Predicted that electrons would emit energy continuously at all wavelengths.
- However, atoms emit light only at specific wavelengths (seen in atomic spectra), which the planetary model could not explain.

Niels Bohr’s Contribution:
- Joined Rutherford’s group in England in 1911.
- Aimed to resolve the issues in the nuclear model by incorporating Planck’s concept of quantized energy levels and Einstein’s theory of light

Stable Condition and Stationary States

Bohr proposed that the laws of electromagnetism do not apply inside the atom.

An electron in a stable orbit in an atom does not radiate energy, despite accelerating.

This stable condition is called a stationary state, with each state having its own specific amount of energy.

Bohr postulated that the energy levels in an atom are quantized, meaning they exist in specific, discrete amounts.

Quantization of Energies

The quantization of energies of electrons in atoms can be likened to a flight of stairs with decreasing-height steps.

Electrons have quantized amounts of energy, each called an energy level.

An electron's energy cannot have a value between allowed energy levels.

The smallest allowable amount of energy is the ground state, while any level above it is an excited state.

Energy of an Atom:

The total energy of an atom is the sum of the kinetic energy of the electrons and the potential energy due to the attraction between electrons and the nucleus.

Electrons closer to the nucleus have lower energy, while electrons further away have higher energy (excited states).

Moving an electron farther from the nucleus requires work, so atoms with electrons in distant orbits have higher energy.

Quantization of Energy and Orbits:
- Because energy is quantized, the electron orbits (distances from the nucleus) are also quantized.
- This concept forms the basis of Bohr’s atomic model, where electrons orbit the nucleus in fixed, quantized energy levels.

Bohr’s Explanation of Atomic Emission Spectra:
- Bohr proposed that stable atoms do not radiate energy while in a fixed orbit.
- Electromagnetic energy is emitted only when an electron transitions from a higher energy level (excited state) to a lower energy level.
- Using Einstein's photoelectric theory, Bohr determined that the energy of each emitted photon is EPhoton=hf, where h is Planck’s constant and f is the frequency of the photon.
- When an atom absorbs a photon, it gains energy equal to that of the photon, and when it returns to a lower energy state, it emits a photon with a characteristic energy

Energy Change in Atomic Transitions:

When an atom transitions from an initial energy level Ei to a final energy level Ef, the change in energy (ΔEatom) is calculated by:
ΔEatom=Ef−Ei

If the energy decreases (a transition to a lower energy level), the energy difference corresponds to the energy of the emitted photon.

Photon Energy Equation:

The energy of the emitted photon Ephoton is equal to the negative of the change in the atom’s energy:
Ephoton=−ΔEatom

Alternatively, it can be expressed as:
Ephoton=Ei−Ef

Planck’s Equation for Photon Energy:
- The energy of the emitted photon is also given by Planck’s equation:
  Ephoton=hf
- Here, h is Planck’s constant, and f is the frequency of the emitted photon.
- Therefore, the energy of an emitted photon is both the product of Planck’s constant and frequency, as well as the decrease in the atom’s energy.

Predictions of the Bohr Model

Predictive Power of a Theory:
- Bohr’s model provided predictions that could be tested experimentally, a key aspect of a strong scientific theory.
- A successful theory should not only explain observed phenomena but also make accurate predictions.

Application to Hydrogen Spectrum:
- Bohr’s model accurately predicted the wavelengths of light emitted by a hydrogen atom, matching experimental measurements by other scientists.
- This agreement between prediction and experiment led to widespread acceptance of Bohr's model for hydrogen.

Limitations of the Bohr Model:
- The model only worked for hydrogen and couldn’t predict the spectrum of helium, the next simplest element.
- It also lacked a strong explanation for why electromagnetic laws should work everywhere except within the atom itself.

Development of Bohr’s Model

Application of Newton’s Second Law and Coulomb’s Law:
- Bohr used Newton’s second law of motion (F=ma) and Coulomb’s law to describe the electron’s motion around the nucleus.
- Coulomb’s law for the force F between an electron (charge −e) and a proton (charge +e) separated by distance r is given by:
  F=−Ke^2/r^2
- The centripetal acceleration of the electron is a=r/v^2, where v is the electron’s velocity.

Quantization of Angular Momentum:
- Bohr proposed that the electron’s angular momentum is quantized, meaning it can only have certain discrete values.
- The quantization condition is:
  
  mevr=nh/2π
- Here, me is the electron’s mass, r is the radius of the orbit, h is Planck’s constant, and n is an integer (the principal quantum number).

Relationship for Orbital Radius:
- By combining the quantization of angular momentum and Coulomb’s force, Bohr derived an expression for the radius of the electron’s orbit for different energy levels in a hydrogen atom.

ELECTRON ORBITAL IN HYDROGEN :

rn= h^2(n^2)/4pi^2Kme(e^2)

Bohr's Model of the Hydrogen Atom

Quantization of Angular Momentum:
- According to Bohr's model, the angular momentum of an electron orbiting the nucleus is quantized and is given by the equation
  
  mevr=nh/2π
  
  where:
  - me = mass of the electron (9.11×10−31kg)
    
    9.11×10−31 kg9.11 \times 10^{-31} , \text{kg}
  - v = velocity of the electron
  - r = radius of the orbit
  - n = principal quantum number (integer values: 1, 2, 3, ...)
  - h = Planck's constant (6.626×10−34J⋅s)

Coulomb's Law (Force between Electron and Nucleus):
- The centripetal force keeping the electron in orbit is provided by the electrostatic attraction between the positively charged nucleus and the negatively charged electron:
- F = Ke^2/r^2
  
  where:
  - K = Coulomb's constant (9.0×109N⋅m2/C2)
  - e = charge of the electron (1.60×10−19C)

Relationship Between Centripetal Force and Coulomb's Force:
- Equating the centripetal force to the Coulomb force:
  
  mev2r=Ke2r2

Deriving the Orbital Radius:
- Using Bohr's quantization of angular momentum and the above relationship, we can solve for the orbital radius
  
  rn=n2h24π2Kmee2
- Substituting known values for constants and n=1n = 1n=1 (for the ground state)
  
  r1=(6.626×10−34 J⋅s)2×124π2(9.0×109 N⋅m2/C2)(9.11×10−31 kg)(1.60×10−19 C)2r_1 = \frac{(6.626 \times 10^{-34} , \text{J} \cdot \text{s})^2 \times 1^2}{4 \pi^2 (9.0 \times 10^9 , \text{N} \cdot \text{m}^2/\text{C}^2) (9.11 \times 10^{-31} , \text{kg}) (1.60 \times 10^{-19} , \text{C})^2}
  
  r1=5.3×10−11 m or 0.053 nmr
- This is known as the Bohr radius.

Calculating the Total Energy of the Electron:
- The total energy of the electron in orbit is the sum of its kinetic energy (KE) and potential energy (PE)
  
  E=KE+PE
- The kinetic energy is given by
KE= 1/2mv^2
- The potential energy due to electrostatic attraction is:
PE=-Ke^2/r
- Using KE=−1/2PE the total energy E is :
E=-Ke^2/2r

Energy Levels of the Hydrogen Atom:
- Substituting the expression for rnr_nrn into the equation for energy, we get :
  
  En=−Ke^4me/2h^2 x 1/n^2
- Substituting numerical values:
  
  En=−2.17×10−18 J×1/n2
- Converting the energy from joules to electron volts (1 eV=1.6×10−19 J):
En=−13.6eV×n21

Summary

Bohr Radius (n=1n ): r1=0.053 nm

Energy Levels:

En=−13.6 eV×1n2
- The energy is quantized and is inversely proportional to the square of the principal quantum number n.
- The lowest energy state (n=1) has an energy of −13.6eV, and as n increases, the energy levels get closer together.

Visual Representation

For n=1,2,3,…:
- Radius increases as rn∝n2.
  
  rn∝n2r_n \propto n^2
- Energy decreases in magnitude as En∝−1/n2
  
  En∝−1/n2E_n \propto -1/n^2

Why is the Energy Negative in the Bohr Model?

In the Bohr model, the total energy of an electron in orbit around a nucleus is negative. This negative sign indicates that the electron is in a bound state, meaning it is attracted to the nucleus and requires energy to be freed from this attraction.

Understanding Zero Energy:

The zero energy level is defined as the point where the electron is infinitely far from the nucleus, having no kinetic energy (i.e., it is at rest). In this state, the atom is ionized, and there is no interaction between the nucleus and the electron.

Therefore, any electron bound to the nucleus (i.e., in any orbit) has a total energy less than zero. The energy is negative because it represents the energy required to pull the electron from its orbit (bound state) to a state of zero interaction (ionization).

Ionization Energy:

Ionization energy is the amount of energy needed to completely remove the electron from the atom, taking it from a bound state to a free state.

For a hydrogen atom, Bohr's model calculates the ionization energy as the energy difference between the ground state (n=1) and the ionized state (n→∞).
E1=−13.6eV
Thus, to ionize a hydrogen atom from its ground state, 13.6 eV of energy is required.

Energy Levels and Spectral Lines:

The energy levels in a hydrogen atom are quantized and given by the formula:
En=−13.6eV×1/n^2
where n is the principal quantum number.

When an electron transitions between these energy levels, it either absorbs or emits a photon with an energy equal to the difference between the initial and final states:
ΔE=Efinal−Einitial

Visible, Ultraviolet, and Infrared Spectral Lines:

The hydrogen atom can emit electromagnetic radiation in different regions of the spectrum:
- Ultraviolet light is emitted when the electron falls to the ground state (n=1) from a higher energy level. This set of transitions is known as the Lyman series.
- Visible light is emitted when the electron falls to the n=2 level from a higher level (n≥3). This set of transitions is known as the Balmer series.
  - The four visible lines in the hydrogen spectrum (red, blue-green, blue-violet, and violet) correspond to transitions from n=3,4,5,and6→n=2, respectively.
- Infrared light is emitted when the electron falls to the n=3 or higher levels (e.g., n=4,5,…). This set of transitions is known as the Paschen series (for n=3) and others.

The Significance of the Bohr Model:

Even though the Bohr model could not accurately predict the spectral lines of atoms with more than one electron, it was crucial in advancing the understanding of atomic structure.

Contributions of the Bohr Model:
- It accurately predicted the emission spectrum of hydrogen, including the precise wavelengths of its spectral lines.
- It provided a method to calculate the ionization energy of a hydrogen atom, which matched experimental data closely.
- The model introduced the concept of quantized electron orbits, which laid the groundwork for quantum mechanics.
- Bohr’s insights into electron arrangements also influenced our understanding of chemical bonding and the periodic table, emphasizing that each element has a unique electron configuration.

Historical Impact:

Niels Bohr received the Nobel Prize in Physics in 1922 for his work on the structure of atoms and their radiation, marking a major step in the development of modern atomic theory.

Summarize result (100%)

The Nuclear Model

Subtopic 2: Rutherford's Experiment

Subtopic 3: Alpha Particles

Surprising Results

Spectroscopy and Emission Spectrum Analysis

Summary

Visible, Ultraviolet, and Infrared Spectral Lines:

Stable Condition and Stationary States

Key Concept:

Bohr's Model of the Hydrogen Atom

Why is the Energy Negative in the Bohr Model?### Energy Levels and Spectral Lines:

Original text

The Nuclear Model

Subtopic 1: Thomson's Model

Subtopic 2: Rutherford's Experiment

Subtopic 3: Alpha Particles

Subtopic 4: Rutherford's Conclusion

Surprising Results

Unexpected Observations

Revised Model

Emission Spectra: Understanding Atomic Arrangement

Gas-Discharge Tubes and Emission Spectra

Spectroscopy and Emission Spectrum Analysis

NOTE :

Emission Spectra

Unknown Samples

Absorption Spectra

Key Concept:

Practice & Application

NOTE

Final Summary:

Observation of Fraunhofer Lines

Determining Star Composition

Importance of Spectroscopy

Revising Rutherford’s Nuclear model

Stable Condition and Stationary States

Quantization of Energies

Energy of an Atom:

Predictions of the Bohr Model

Development of Bohr’s Model

Bohr's Model of the Hydrogen Atom

Summary

Visual Representation

Why is the Energy Negative in the Bohr Model?

Understanding Zero Energy:

Ionization Energy:

Energy Levels and Spectral Lines:

Visible, Ultraviolet, and Infrared Spectral Lines:

The Significance of the Bohr Model:

Historical Impact:

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