Showing posts with label Atoms. Show all posts
Showing posts with label Atoms. Show all posts

Problems and Solutions on Bohr's Atomic Model

Problem and solution

When the electron in hydrogen atom jumps from second orbit to first orbit, a certain wavelength is emitted. When the electron jumps from the third orbit to first orbit, what is the new wavelength emitted in terms of the first wavelength?

We can solve this problem basing on the derivation is that we made for the reciprocal of the wavelength in terms of the Redberg constant. By applying the given condition in the problem in two different equations and by simplifying them we can solve the problem as shown below.




Problem and solution

What is the ratio of largest to shortest wavelengths in the Balmer series of the hydrogen spectrum?

We can solve this problem also basing on the derivation that we made for the reciprocal of the wavelength in terms of principal quantum numbers. For the wavelength to be maximum, the corresponding energy has to be minimum. It is possible only when the electronic jumping from the third orbit to second orbit.

Further wavelength to be minimum, the corresponding energy has to be maximum. This is possible when their electron is jumping from infinite orbit to second orbit. The corresponding equation is written in the problem is solved as shown below.




Problem and solution

In a hydrogen atom electron is jumped from the fifth orbit to first orbit. What is the recoil speed of the hydrogen atom in this process?

As the electronic jumping from higher orbit to lawyer orbit, there is some emission of energy. This emitted energy will have a certain wavelength. To compensate the jerk that is generated by this emitted energy, nuclear is most with a little bit velocity and here we are going to calculate that velocity. By substituting the wavelength that we have derived in the de Broglie concept we can derive the equation for the velocity of the nucleus as shown below.




Problem and solution

If the wavelength of the first member of the Balmer series in the hydrogen spectrum is 6564 Å, what will be the wavelength of the second member of the Balmer series?

We can solve this problem by writing the equation for the reciprocal of the wavelength using the atomic model. By comparing the given two conditions we can get the answer as shown below.



Problem and solution





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De Broglie’s explanation for Bohr’s Angular Momentum

As per the Bohr’s atomic model, the electron in a specified orbit will have a constant angular momentum. This orbit is called stationery orbit and here de Broglie proved that electron in this orbit is going to have a constant angular momentum.

According to de Broglie concept all material particles like electrons have wave nature. So the electron travels like a wave along the length of the circumference of the orbit. The stationary waves can be formed only when the wavelength are equal to circumference of the circular orbit or its integral multiples.

The expression is as shown below.



Though de Broglie is able to explain it successfully, in the process of explanation he assumed electron like a wave which is against the concept of Planck’s constant theory.

Bohr’s atomic model is valid only for the hydrogen and hydrogen like atoms that are having only one electron. If there is more number of electrons, between them there is a repulsive force which is not taken into consideration during this Bohr’s atomic model.

Therefore some more theories are proposed to explain the atomic nature of the atoms.





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Atoms Complete Lesson

In this chapter Atoms,we are going to discuss about distribution of electron in the orbit and its corresponding time period,velocity,potential energy,kinetic energy,wave length and frequency.We also learn about uncertainty principle. All the relevant posts are linked here in this page.








Other Complete Lessons in this blog includes 

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Bohr's Line spectrum of hydrogen atom

When an electron jumps from higher orbit to lower orbit, the difference in the energies of the orbits is released in the form of photon with a certain frequency. By writing the equations for their energies and by subtracting them we can derive the equation for the frequency and the wavelength as shown below. Here we will get a constant called as Ridberg constant.



Emission spectrum of hydrogen atom

Electron in a hydrogen atom can be navigated straight for a very small time. It is not stable there therefore it always preferred to jump into the lower orbits. Once if it is in the lower orbit it will have less potential energy and it will be more stable. Therefore electrons always tended to jump from higher orbits to lower orbits whenever there is a vacancy. The emitted energy in this process is in the form of photons. This photon will have a certain frequency and the spectrum of all these frequencies could be called as emission spectrum.



Lyman series

Depending on electron jumping from which orbit to which orbit, the emitted energies and their corresponding frequencies are different. If the electronic jumping into the ground state, the corresponding spectral line is called Lyman series. In this case, electrons from different higher orbits are always going to jump into the ground state. We can calculate the corresponding wavelength and frequencies and they are maximum and minimum values as shown below. This region in the ultraviolet region which is in invisible region.



Balmer series

If the electrons from the different orbits jump into the second orbit, the corresponding emitted energies in the frequencies are called Balmer series. These corresponding wavelengths and frequencies are in the visible region. In fact this is the only frequencies that are visible in the bourse atomic model.

We can derive the corresponding wavelength and frequencies for this region as shown below.



Passion series and Bracket series

If the electron is jumping into the third orbit the corresponding frequencies emitted are represented as Passion series. If the electron jumps into the fourth orbit from different orbits the corresponding frequencies set is called Bracket series. Their corresponding wavelengths and energies are as shown below. The respective spectrum diagram is also shown.



We can calculate the number of the spectral lines possible between two specified orbits as shown below.



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Potential and Kinetic energy of electron in the orbit

We know that the electron is revolving in a specified orbit. We also know that these orbits are called stationery orbits. When the electron is in the stationary orbit, it neither loose the energy nor gain the energy. Hence it has a fixed energy. This energy is both in the forms of potential energy and kinetic energy. We can derive the equations for both potential and kinetic energies as shown below. By adding potential and kinetic energies we can get the total energy of the electron. It is the observed by the results that the total energy and the potential energy are negative values. It is a symbolic way of expressing that the electron is associated with the nucleus.

It is also noticed that potential energy is twice the magnitude of total energy. It is observed that the magnitude of total energy and the kinetic energy are equal.

The expression for the kinetic energy is done taking the basic equation of kinetic energy and substituting the value of the velocity of the electron that is derived. Again potential energy is defined basing on the very definition of electrostatic potential energy. By adding both these energies we got total energy.

It is observed that all these energies are directly proportional to Squire of the atomic number and inversely proportional to Squire of principal quantum number.



Energy levels of electron in different orbits

We know that electrons revolve is in different orbits around the nucleus. Basing on the orbit each electron will have different energy. As the number of the orbit is increasing the total energy of the electron also increases and at the infinite orbit it’ll becomes zero. We can express the total energy of any of the electron in terms of the basic energy of the electron in the ground state which is equal to -13.6 electron volt.

Energy of the electron in any other orbit is this numerical value divided by Squire of the principal quantum number. Basing on the above result, we can write energies of different orbits not only for the hydrogen atom, even for the hydrogen like attempts as shown below.



Problem and solution

An electron in a hydrogen atom makes the transition from one orbit to another orbit. Assume that the Bohr atomic model is valid. It is given that the time period of the electron in the initial state is eight times of the final state. What are the possible values of principal quantum numbers?

We have proved that the time period of an electron in a given orbit is directly proportional to cube of principal quantum number. Taking this into consideration we can derive a direct relation between principal quantum numbers. As we know that the principal quantum number is not a fractional value, basing on that and the relation we can write different sets of values of principal quantum numbers for the above problem as shown below.




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Velocity and Time period of electron in Bohr orbit

Velocity of the electron in the orbit

We know that according to Bohr’s atomic model, electrons revolve in specified orbits. These orbits are called stationery orbits. When the electron is in that orbit, it neither loose energy nor gain that energy. The electron in this orbit will have a specific velocity and we can derive the equation for the velocity.

According to second postulate the angular momentum of the electron is integral multiples of a constant. This number which is an integral multiple is actually called as principal quantum number. Taking this concept into consideration we can write the equation for the velocity of the electron in that terms. It can be mathematically proved that velocity of the electron in any orbit is independent of mass of electron. It is directly proportional to atomic number and is inversely proportional to principal quantum number. We can calculate the velocity of the electron in the first orbit by writing atomic number as well as the principal quantum member equal to 1. The expression for the velocity is derived as shown below.



Time period of the electron in the orbit

As the electron is revolving the in a circular path, it take specific time to complete one rotation. This specific time is called time period. We can derive the equation for the time period by writing a small relation between linear velocity and the angular velocity of the electron. The derivation is as shown below. It is proved that time period of the electron is directly proportional to cube of the principal quantum number and inversely proportional to Squire of the atomic number.




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Bohr’s atomic model and Radius of Orbit

As Rutherford Alfa scattering model failed to explain all the properties of the matter, a new concept called Bohr’s atomic model is introduced. This atomic model is valid for hydrogen atom and the hydrogen like atoms. Plank’s quantum concept is taken into consideration to explain this atomic model. This model has some basic postulates.

According to Bohr’s atomic model, electron can revolve around the nucleus only in specified orbits. These orbits are called stationery orbits. When the electron is revolving in this orbit, it neither loses the energy nor gains the energy. These orbits are circular in nature. 

For the electron to take that circular path it needs some centripetal force. Centripetal force is never an outside force. The force that is acting in the system towards the Centre of the system is called centripetal force. In this case the electric force of attraction between the electron and the protons of the nucleus provides the necessary centripetal force.

According to second postulate, the revolving electron has a constant angular momentum which can be expressed in terms of Planck’s constant.

According to third Bohr’s postulate, energy of the electron in the given orbit is constant. It poses both potential and kinetic energies. The sum of these total energies is always constant and the electron is not going to radiate any energy when it is orbiting in a stationary orbit.

When the electron jumps from the higher orbit to lower orbit, it releases energy. If the electron has to jump from the lower orbit to higher orbit it need energy. This energy is either emitted or absorbed in the form of wave packets called quanta. This energy will have a certain frequency and wavelength. The energy emitted or absorbed is in the integral multiples of product of Planck’s constant and the emitted frequency.



Determination of the radius of Bohr orbit

According to Bohr’s concept, the electron is revolving in a specified orbit around the nucleus. These orbits are called stationery orbits and when the electron is in this orbit it neither loose the energy nor gain the energy. For the electron to continue in the circular path it need some centripetal force. This centripetal force is provided by the electric force of attraction between the electron and the protons of the nucleus.

The electron in a circular path will always have a constant angular momentum. Taking these two things into consideration we can derive the equation for the orbital of an electron as shown below.




It can be noticed that the radius of the electron depends on the principal quantum number, mass of the electron and atomic number. It can be observed that radius is directly proportional to Squire of the principal quantum number, inversely proportional to both mass and atomic number.

For the hydrogen atom we can substitute atomic number is equal to 1. As all the terms of the equation of the radius ground state orbit are constant, we can calculate the radius of the first orbit and it can be mathematically shown that it is numerically equal to 
0.53 Å. This value is called Bohr’s radius and the corresponding orbit is called Bohr’s orbit.


Taking this value into consideration we can write the numerical value of the further orbits as shown in the above diagram.


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Rutherford Alfa experiment and observations about Atom

Physics is a branch of science where we study nature. It deals with different properties of matter. We know that the matter has three different states like solids state, liquid state and gaseous state. All the states the matter consists of molecules. It is understood that molecules can be divided into atoms. When they were named as atoms, it means they are not divisible. But later we are able to divide atoms further into nucleus and the electrons which are revolving around the nucleus.

To know the structure of atoms, we have different theories and the corresponding experimental results.

We are having a model called pulp pudding model. It is also called watermelon model. According to this theory electrons and protons are uniformly distributed in the nucleus like the seeds of a watermelon. As this theory is unable to explain all the properties of matter, a new theory called Rutherford Alfa experimental model is introduced.

According to Rutherford Alfa experiment model nucleus is consisting of positive charges and the size of the nuclei is much smaller than that of the atom. To verify this, an experiment is performed. In this experiment helium particles are allowed to strike a thin gold foil. Helium particles consist of positive charge. As the gold foil is very thin, we can assume that helium particles can directly go and interact with the atom itself.

It is experimentally observed that, out of 8000 alfa rays that were driven towards the nucleus, only one is deviated. From that it can be concluded that the nucleus is also having a positive charge. Then only there will be repulsion and the alpha ray and nucleus so that path is deviated. It can also be concluded that the size of the nucleus is very small when compared with the size of the atom. That is why out of so many positive rays only very few are reflected.

As there is repulsion between the two positive charges, alpha particles cannot go and strike the nucleus. After they reach a certain distance, the repulsion dominates them so that either they deviates or reflects back. That’s why the particles are able to reach up to only a certain distance from the nucleus and this distance is called closest distance of approach. At that point all the kinetic energy of the alpha particle is converted into electrostatic potential energy between the Alpha particle and the protons of the nucleus.





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