Concept explainers
(a)
The
Answer to Problem 77QAP
Mathematically the angular momentum of an electron can be written as
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
According to Bohr quantization principle angular momentum of an electron in an atom is an integral multiple of
So mathematically we can write it as,
(1)
where, m is the mass of the electron, vnis the velocity on the nthorbit, n is a positive integer.
Conclusion:
So mathematically the angular momentum of an electron can be written as
(b)
The radius of the electron orbit in hydrogen atom
Answer to Problem 77QAP
The radius of the electron orbit in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
Electrostatic force between the orbiting electron and proton supplies the required
(2)
where
(3)
Now using equation (1) in equation (3) we can write,
(4)
Conclusion:
So, the radius of the nthelectron orbit in hydrogen atom is given by
(c)
The kinetic energy of an electron in hydrogen atom
Answer to Problem 77QAP
The kinetic energy of an electron in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
Kinetic energy of the electron is given by
(5)
Now substituting the expression for rnin equation (1) we obtain
(6)
Now substituting the value of vnin equation (5) we obtain
(7)
Conclusion:
So, the kinetic energy of an electron in hydrogen atom is given by
(d)
The total energy of an electron in hydrogen atom
Answer to Problem 77QAP
The total energy of an electron in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
Total energy of the electron is given by the sum of kinetic and potential energy
(8)
Knexpression we have already derived in part (c). Now we want to derive an expression for the potential energy Epwhich is the electro static energy between proton and the orbiting electron. So
(9)
Now substituting the value of rnusing equation (4) in equation (9) we get
(10)
Now putting equation (7) and (10) in equation (8) we get
Conclusion:
So, the total energy of an electron in hydrogen atom is given by
(e)
The speed of an electron in hydrogen atom
Answer to Problem 77QAP
The speed of an electron in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
We have already derived the expression for speed of an electron in part (c) equation (6). The expression for speed in the nth orbit of the electron is given by
Conclusion:
So, the speed of an electron in hydrogen atom is given by
Want to see more full solutions like this?
Chapter 26 Solutions
COLLEGE PHYSICS
- An atom can be formed when a negative muon is captured by a proton. The muon has the same charge as the electron and a mass 207 times that of the electron. Calculate the frequency of the photon emitted when this atom makes the transition from n=2 to the n=1 state. Assume that the muon is orbiting a stationary proton.arrow_forwardConstruct Your Own Problem The solar corona is so hot that most atoms in it are ionized. Consider a hydrogen-like atom in the corona that has only a single electron. Construct a problem in which you calculate selected spectral energies and wavelengths of the Lyman, Balmer, or other series of this atom that could be used to identify its presence in a very hot gas. You will need to choose the atomic number of the atom, identify the element, and choose which spectral lines to consider.arrow_forwardConstruct Your Own Problem Consider the Doppler-shifted hydrogen spectrum received from a rapidly receding galaxy. Construct a problem in which you calculate the energies of selected spectral lines in the Balmer series and examine whether they can be described with a formula like that in the equation 1=R(1nf21ni2), but with a different constant R.arrow_forward
- Verify that the ground state energy E0 is 13.6 eV by using E0=22qe4mek2h2.arrow_forwardIntegrated Concepts In a Millikan oil-drop experiment using a setup like that in Figure 30.9, a 500-V potential difference is applied to plates separated by 2.50 cm. (a) What is the mass of an oil drop having two extra electrons that is suspended motionless by the field between the plates? (b) What is the diameter of the drop, assuming it is a sphere with the density of olive oil?arrow_forwardA hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is n for a hydrogen atom if 0.850 eV of energy can ionize it?arrow_forward
- JC-41) Angular Momentum in the Hydrogen Atom An electron is in an angular momentum state with e 3. a) What is the length of the electron's angular momentum vector? b) How many different possible z components can the angular momentum vector have? c) What are the values of the angle that the L vector makes with the z-axis? Make a table of your answers. Express your answer in degrees.arrow_forward• Which of the following postulates of the Bohr model led to the quantization of energy of the hydrogen atom? A- The electron goes around the nucleus in circular orbits. B- The angular momentum of the electron can only be an integral multiple of h/2a. C- The magnitude of the linear momentum of the electron is quantized. D- Quantization of energy is itself a postulate of the Bohr model.A. Electrons, • The Bohr model of atoms A- assumes that the angular momentum of electrons is quantized. B- uses Einstein's photoelectric equation. C- predicts continuous emission spectra for atoms. D- predicts the same emission spectra for all types of atoms. • The diagram shows the energy levels for an electron in a certain atom. Which transition shown represents the emission of a photon with the most energy? R= 2 =1- %3D IV A-I A-I В-П C- III D- IV • The Bohr model of an atom A- assumes that the angular momentum of electrons is quantised. B- uses Einstein's photoelectric equation. C- predicts continuous…arrow_forward3) In Millikan’s oil drop experiment an oil drop with N additional electrons and mass m is subjected to an electric field E. It is observed that the oil drop is moving downward at constant velocity v. The friction force on the electron is kv. Gravity is negligible. a) Draw a figure and show the directions of the friction force Ff and the electric force Fe. b) Find the charge q of the oil drop in terms of k, v and E. c) Find the number of electrons N on the oil drop in terms of k,v, E and e.arrow_forward
- • Using the Bohr model, estimate the energy of a Ka X-ray emittedby lead 1Z = 822arrow_forward8. A white dwarf is a very dense star, with its ions and electrons packed extremely close together. Each electron may be considered to be located within a region of size x= 1.5 x 10-12 m. Estimate the minimum speed of the electron. Do you think that the effects of relativity will be important for these stars?arrow_forward45-46. Find the increase in energy of a hydrogen atom from orbit n = 1 to orbit n = 3. • 12.08 eV • 12.08 eV • -13.6 eV • 13.6 eVarrow_forward
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningCollege PhysicsPhysicsISBN:9781938168000Author:Paul Peter Urone, Roger HinrichsPublisher:OpenStax CollegeCollege PhysicsPhysicsISBN:9781285737027Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
- University Physics Volume 3PhysicsISBN:9781938168185Author:William Moebs, Jeff SannyPublisher:OpenStax