Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

A hypothetical atom has energy levels uniformly separated by 1.2 eV . At a temperature of 2000 K, what is the ratio of the number of atoms in the $13^{th}$ excited state to the number in the $11^{th}$ excited state?

The ratio of the number of atoms in the $13^{th}$ excited state to the number of atoms in the $11^{th}$ excited state is $9 \times 10^{- 7}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data:

The energy difference between two atomic levels of an atom, $∆ E = 1.2 eV$

The temperature of the states, $T = 2000 K$

Number of atoms in the higher-energy state, $N_{1} = 6.1 \times 10^{13} / {cm}^{3}$

Number of atoms in the lower-energy state, $N_{2} = 2.5 \times 10^{15} / {cm}^{3}$

Step 2: Understanding the concept of Boltzmann distribution equation

The Boltzmann distribution is a probability function used in statistical physics to define the state of a particle system in terms of temperature and energy.

Using the Boltzmann-distribution equation which is the expression for the probability for stimulated emission of radiation to the probability for spontaneous emission of radiation under thermal equilibrium, we can get the required ratio of the number of atoms present in the excited state to that present in the excited state.

Formula:

The Boltzmann energy distribution equation,

$\frac{N_{1}}{N_{2}} = e^{- (E_{2} - E_{2}) / kT}$ ….. (1)

Here, Boltzmann constant, $k = 8.625 \times 10^{- 5} eV / K$

Step 3: Calculation of the ratio of number of atoms in the given states:

The energy difference between any two levels is,

$E_{13} - E_{11} = 2 (1.2 eV) = 2.4 eV$

So, the energy difference between the 13^th excited state and the 11^th excited state is given as:

Now, using this energy difference value in the equation (i) with the given data, we can get the ratio of the number of atoms in the 13^th excited state to the number of atoms in the 11^th excited state as follows:

$\frac{N_{1}}{N_{2}} = e^{- \frac{(2.4 eV)}{(8.625 \times 10^{- 5} eV / K) (2000 K)}} = e^{- 13.91} = 9 \times 10^{- 7}$

Hence, the value of the required ratio is $9 \times 10^{- 7}$ .

Element	λ (pm)	Element	λ (pm)
Ti	275	Co	179
V	250	Ni	166
Cr	229	Cu	154
Mn	210	Zn	143
Fe	193	Ga	134

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Fundamentals Of Physics

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Short Answer

A hypothetical atom has energy levels uniformly separated by 1.2 eV . At a temperature of 2000 K, what is the ratio of the number of atoms in the $13^{th}$ excited state to the number in the $11^{th}$ excited state?

Step by Step Solution

Step 1: The given data:

Step 2: Understanding the concept of Boltzmann distribution equation

Step 3: Calculation of the ratio of number of atoms in the given states:

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Fundamentals Of Physics

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Short Answer

A hypothetical atom has energy levels uniformly separated by 1.2 eV . At a temperature of 2000 K, what is the ratio of the number of atoms in the 13thexcited state to the number in the 11th excited state?

Step by Step Solution

Step 1: The given data:

Step 2: Understanding the concept of Boltzmann distribution equation

Step 3: Calculation of the ratio of number of atoms in the given states:

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