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College Physics (Urone)
Found in: Page 1112
College Physics (Urone)

College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

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Short Answer

Find the radius of a hydrogen atom in the n = 2 state according to Bohr’s theory.

The radius of a hydrogen atom in the 2nd state is \[2.116 \times {10^{ - 10}}\;{\rm{m }}{\rm{.}}\]

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determine the Bohr’s theory

The spectrum of hydrogen atoms is explained by an atomic structure theory. It is assumed that the electron orbiting the nucleus can only exist in particular energy states, with each transition resulting in the emission or absorption of a quantum of radiation.

Step 2: Given information and Formula to be used

Consider the formula for the radius of the atomic orbit is as follows:

\[{r_B} = \frac{{{n^2}{a_B}}}{z}\]

Here, rB is the radius of orbit, n is the energy level, aB is the Bohr's radius and z is the number of proton.

Step 3: Determine the radius of a hydrogen atom in the 2 state. 

Radius of hydrogen atom is calculated as

\[{r_B} = \frac{{{n^2}{a_B}}}{z}\]

Substitute the value in the above expression

\[\begin{array}{c}{r_B} = \frac{{{{\left( 2 \right)}^2}\left( {0.529 \times {{10}^{ - 10}}\;{\rm{m}}} \right)}}{1}\\{r_B} = {\left( 2 \right)^2}\left( {0.529 \times {{10}^{ - 10}}\;{\rm{m}}} \right)\\{r_B} = 4\left( {0.529 \times {{10}^{ - 10}}\;{\rm{m}}} \right)\\{r_B} = 2.116 \times {10^{ - 10}}\;{\rm{m}}\end{array}\]

Hence, the radius of a hydrogen atom in 2 state is \[2.116 \times {10^{ - 10}}\;{\rm{m}}\].

Most popular questions for Physics Textbooks

Calculate the mass of a proton using the charge-to-mass ratio given for it in this chapter and its known charge.

(b) How does your result compare with the proton mass given in this chapter?

(a) what voltage must be applied to an X-ray tube to obtain 0.0100 fm wavelength X-rays for use in exploring the details of nuclei? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) If one subshell of an atom has 9 electrons in it, what is the minimum value of l ?

(b) What is the spectroscopic notation for this atom, if this subshell is part of the n = 3 shell?

Particles called muons exist in cosmic rays and can be created in particle accelerators. Muons are very similar to electrons, having the same charge and spin, but they have a mass 207 times greater. When muons are captured by an atom, they orbit just like an electron but with a smaller radius, since the mass in \({{\bf{a}}_{\bf{B}}}{\bf{ = }}\frac{{{{\bf{h}}^{\bf{2}}}}}{{{\bf{4}}{{\bf{\pi }}^{\bf{2}}}{{\bf{m}}_{\bf{e}}}{\bf{kq}}_{\bf{e}}^{\bf{2}}}}{\bf{ = 0}}{\bf{.529 \times 1}}{{\bf{0}}^{{\bf{ - 10}}}}{\bf{\;m is 207}}{{\bf{m}}_{\bf{e}}}\)

(a) Calculate the radius of the n = 1 orbit for a muon in a uranium ion\(\left (Z = 92).

(b) Compare this with the 7.5 - fm radius of a uranium nucleus. Note that since the muon orbits inside the electron, it falls into a hydrogen-like orbit. Since your answer is less than the radius of the nucleus, you can see that the photons emitted as the muon falls into its lowest orbit can give information about the nucleus.

What two pieces of evidence allowed the first calculation of me, the mass of the electron?

(a) The ratios \[\frac{{{{\bf{q}}_{\bf{e}}}}}{{{{\bf{m}}_{\bf{e}}}}}\]and \[\frac{{{{\bf{q}}_{\bf{p}}}}}{{{{\bf{m}}_{\bf{p}}}}}\].

(b) The values of qe and EB.

(c) The ratio \[\frac{{{{\bf{q}}_{\bf{e}}}}}{{{{\bf{m}}_{\bf{e}}}}}\]and qe.

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