Bohr's Model of Hydrogen Atom

Bohr's Model of Hydrogen Atom

History and Introduction

• Early scientists aimed to understand the structure of an atom.
• Initial models were simplistic, e.g., the watermelon model with electrons randomly distributed.
• Bohr's model introduced specific orbits for electrons, contrasting with the idea that electrons could revolve anywhere.

Bohr's Postulates

• Electrons revolve in fixed orbits.
• Each orbit has a specific energy level.

Why a Model?

• The exact nature of an atom is still unknown; new discoveries are continuously made.
• Bohr's model is a representation of Bohr's ideas, not necessarily the absolute reality.
• Some aspects of Bohr's model are accurate, while others are not.

Validity of Bohr's Model

• Bohr's model applies only to hydrogen atoms or hydrogen-like atoms/species.
• A hydrogen-like species is an atom or ion with only one electron.

Examples of Hydrogen-like Species

• Hydrogen (H): One electron.
• Helium ion (He+): Helium has two electrons, but when one electron is removed, it becomes hydrogen-like.
• Lithium ion (Li2+): Lithium has three electrons; removing two leaves it with one electron.
• Sodium ion (Na10+): Sodium has 11 electrons; removing 10 leaves it with one electron.

Core Principles of Bohr's Model

• Bohr's theory rests on two main principles:
  • Quantization of angular momentum
  • Force equation

Quantization of Angular Momentum

• Momentum = mass × velocity.
• Angular momentum: The angular effect of momentum; related to rotation.
• Bohr stated that the angular momentum of an electron revolving around the nucleus is quantized.
• Quantization means only specific, fixed values are allowed.
  • The charges that can exist are integral multiples of $e$.
  • Smallest possible charge $e$.
  • Next bigger charge $2e$, then $3e$, $4e$, etc.
  • Between $2e$ and $3e$ no charge is possible.
  • $n \, e$ values only.
• The angular momentum of an electron is quantized, meaning it can only take certain fixed values.
• Minimum value of angular momentum: $\frac{h}{2\pi}$, where $h$ is Planck's constant.
• Possible values: $\frac{h}{2\pi}$, $2 \frac{h}{2\pi}$, $3 \frac{h}{2\pi}$, $4 \frac{h}{2\pi}$, etc.
• Angular momentum (L) formula: $L = mvr = n \frac{h}{2\pi}$
  • $m$ = mass of electron
  • $v$ = velocity of electron
  • $r$ = radius of the orbit
  • $n$ = integer (1, 2, 3, …)
  • $h$ = Planck's constant ($6.6 \times 10^{-34} Js$)

Force Equation

• Circular motion requires a continuous force towards the center (centripetal force) to change direction.
• In Bohr's model, the electron revolves around the nucleus in a circular path, so a force is needed.
• The nucleus (positive charge) attracts the electron (negative charge), providing the necessary centripetal force.
• Coulombic force of attraction between nucleus and electron equals centripetal force.
• Centripetal force: $F_c = \frac{mv^2}{r}$
• Coulombic force: $F<em>e = k \frac{q</em>1 q<em>2}{r^2}$, where $k = \frac{1}{4 \pi \epsilon</em>0}$
  • $q_1$ = charge of the nucleus = $Ze$ ($Z$ is the atomic number)
  • $q_2$ = charge of the electron = $e$
• Force equation: $\frac{mv^2}{r} = \frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r^2}$

Key Formulas from Bohr's Model

• Derived from the principles of quantization of angular momentum and force equation.

Radius of Orbit

• $r = 0.529 \frac{n^2}{Z}$ Ångström
  • $n$ = orbit number
  • $Z$ = atomic number
• 1 Ångström = $10^{-10}$ meters

Velocity of Electron

• $v = \frac{c}{137} \frac{Z}{n}$ m/s
  • $c$ = speed of light ($3 \times 10^8$ m/s)

Energy of Electron

• $E_n = -13.6 \frac{Z^2}{n^2}$ eV
  • eV = electron volt (unit of energy)
    Note: Total energy of electron is negative, indicating attraction to the nucleus. Formula gives total energy, not just kinetic energy.* Kinetic energy is always positive.
• When total energy is negative, it indicates attraction
• To free an electron from the nucleus requires that the electron's total energy is 0 or positive

Energy Components

• Total energy = kinetic energy + potential energy

Kinetic and Potential Energy Relationship to Total Energy

• If total energy is known you can derive each individual energy.

• Total energy: $-2 eV$, the kinetic energy is $+2 eV$ and potential energy is $-4 eV$.

• To get kinetic energy remove the negative sign from the total energy.

• To get the potential energy just multiply the total energy by 2.

• Kinetic = $-$ (Total Energy).

• Potential Equal to $2$ * Total Energy.

• Example Formulas:

  • Kinetic Energy: $+13.6 \frac{Z^2}{n^2} eV$
  • Potential Energy: $-2 * 13.6 \frac{Z^2}{n^2} eV$

Ratios of Kinetic, Potential, and Total Energy

• Important to remember the ratio between these energies

• Important Car analogy when assessing ratio, need to remember car and how to derive.

• Kinetic: Potential Total = $1: -2: -1$

  • Important note: Questions can be asked in multiple ways for total potential and kinetic energy, important to understand this and remember car from analogy.

Derivation of the Formulas

• Two Main Pillars:
  • Quantization of Angular Momentum: $mvr = n \frac{h}{2\pi}$
  • Force Equation: $\frac{mv^2}{r} = \frac{k q<em>1 q</em>2}{r^2}$

Deriving Velocity

• Important relationships, Equation 2 / Equation 1 gives velocity.
• Final formula looks, $\frac{Z e^2}{2 \epsilon_0 h n}$
  • Can represent as $\frac{c}{137} \frac{Z}{n}$
• Key thing is $Z/n$

Deriving Radius

• To derive radius, substitute value of velocity with equation 1.
  • Initial Step $Mvr = \frac{nh}{2 \pi}$
  • Use this and derived equation to figure out Radius.
    • Radius end equation will look as $\epsilon_0 h^2 \frac{n^2}{ \pi m Z e^2}$
  • If put this equation using real constant it looks like $0.529 \frac{n^2}{Z}$

Important Formulas involving derivation:

Formulas to Pay Attention To:

• Pay attention to derivations, derivations are important in exam.
• Look into which formula MASS is present vs in which MASS is not present.
  • Radius Formula mass is present in formula.
    • Radius is independent from mass of electron.
  • If god changed of the mass of electron which quantity will be affected?
    • Velocity not affected, Radius is affected.

How to derive Velocity, Kinetic Energy Using Equation?

• Easy to find using what's known and plugging constants with each other.

  Energy Derivations:

  • Start easy with half $mv^2$
  • Potential energy $k q<em>1 q</em>2/ r$
  • Using that figure easy constants using already known values!

Key Energy Constants:

• Using real derivations one can produce equations that involve different energies to the relationships between values.
  • One being KE with positive signs.
  • Other Potential with negative signs.

Some Important Observations:

• Key point, Velocity is independent to mass!
• Radius is inversely proportional to mass of electron :!
• Different energies are related directly to equation, potential and total energy.
  • If Mass goes up by 5X then radius goes up to be 1/5th, but Mass would remain be consistent.

Hydrogen Calculations:

• Some values should be buttified. (Remember what's important!).
  • Know your total relationships when shell is in first and orbit.

Equation Time - For Hydrogen Atom!

• Equation is $-13.6 \frac{Z^2}{n^2}eV$
  • Z - Atomic Number!
  • Use N from KLMNO.
• Remember equation works for only LIKE Species.

Using above information derive some equations that are useful for problem solving.

• Lets look at some derivations relating with K, for Lithium, Helium +1, as derived.
• You can follow a trend using this model.

Key Point to Remember:

• When electron travel at certain shell , we only total energy, can find for formula.
  • Can find total energy using relationships.

Extra Note: Small the value and more negative: more attractive to the nucleus!

• That's related to the protons!

Last Point of topic. Important: Relationship, Lithium, Sodium and H.

For that is not the relationship for K, relationship goes. K x N from each other.

Diagram Info:

• If you have H diagram can use to show relationships as shown with equations with diagram in notebook and relationships of diagram and how they can show.

Final key of key Important: what extra must to be given Jump to Second Orbit

• Orbit gap of electrons - to travel certain locations in space.
• Relationship of higher number- and using it will require more electron potential.
• Need to provide ENERGY THOUGH Photon with certain levels to reach new level.
  • Story is about pushing electron from number to from one to the another, to know and push electron.
    • Final important: Momenetaum that travel from lower and higher orbit: Does this electron for stay!

Summary of How electron move:

• If dad gives for going to Maldives
  • money - will they for ever stay in the Maldives
    • Short time on Electron - and if in the M-orbit with the energy with for ten, what happens is!
    • Electron - Spiderman from First orbit can take. energy reach third to it.
      *Stay there for for a certain level will be there 8 sec power! Then come BACK.

Side Notes:

• Electron will radiate energy. Gives an emission spectra what spectra emission.
• This gives use atomic spectra.
  • In what happens with Electrons - move from one equation to another and you want to tell what to happen.
  • For JUMP what to do!

End of topic! for continue into the next session! And Good Luck!