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Chapter 10: Problem 23

(a) Find the lowest energy of an electron confined in $$ \text { a box of side } 1 \mathrm{~A} \text { . } $$ (b) Find the temperature at which the average energy of the molecule of the molecule of a perfect gas would be equal to the lowest energy of the electron.

Short Answer

Expert verified
Lowest energy: \(6.023 \times 10^{-18} \text{J}\). Temperature: \(2.91 \times 10^5 \text{K}\).

Step by step solution

01

Determine the formula for the lowest energy

The lowest energy of an electron in a 1-dimensional box is given by the formula: \[ E_n = \frac{n^2 h^2}{8mL^2} \] where n is the principal quantum number, h is Planck's constant, m is the mass of the electron, and L is the length of the box. For the lowest energy, n = 1.
02

Insert known values

Insert the known values for Planck's constant \( h = 6.626 \times 10^{-34} \text{Js} \), the mass of the electron \( m = 9.109 \times 10^{-31} \text{kg} \), and the length of the box \( L = 1 \text{Å} = 1 \times 10^{-10} \text{m} \). Substituting these values into the equation, we get: \[ E_1 = \frac{1^2 (6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (1 \times 10^{-10})^2} \]
03

Calculate the lowest energy

Perform the calculations step-by-step: \[ E_1 = \frac{6.626^2 \times 10^{-68}}{8 \times 9.109 \times 10^{-31} \times 10^{-20}} \] \[ E_1 = \frac{4.39 \times 10^{-67}}{7.287 \times 10^{-50}} \] \[ E_1 \approx 6.023 \times 10^{-18} \text{J} \]
04

Find the temperature corresponding to the lowest energy

The average energy of a molecule in a perfect gas at temperature T is given by the formula: \[ E = \frac{3}{2} k_B T \] where \( k_B \) is the Boltzmann constant \( k_B = 1.38 \times 10^{-23} \text{J/K} \). Setting this equal to the lowest energy of the electron, we can solve for T: \[ E_1 = \frac{3}{2} k_B T \] \[ T = \frac{2E_1}{3k_B} \] \[ T = \frac{2 \times 6.023 \times 10^{-18}}{3 \times 1.38 \times 10^{-23}} \]
05

Calculate the temperature

Complete the calculation: \[ T = \frac{12.046 \times 10^{-18}}{4.14 \times 10^{-23}} \] \[ T \approx 2.91 \times 10^5 \text{K} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

electron energy levels
In quantum mechanics, electrons can only occupy discrete energy levels. These are determined by quantum numbers.
Each energy level represents a state of definite energy.
For an electron in an atom, these energy levels depend on the electron's distance from the nucleus and the principal quantum number, n. The energy levels can be calculated using the expression \(E_n = \frac{n^2 h^2}{8mL^2} \)
For the lowest energy level, n=1 which represents the ground state.
particle in a box
The 'particle in a box' model helps in understanding quantum energy quantization. This simple model confines a particle such as an electron in a box with impenetrable walls.It assumes that the particle cannot exist outside the box.
The allowed energy levels within the box are quantized and can be calculated using the formula \(E_n = \frac{n^2 h^2}{8mL^2} \)
Here, h is Planck's constant, m is the particle's mass, L is the length of the box, and n is an integer called the quantum number.
quantum energy quantization
Quantum energy quantization refers to the discrete energy levels within which quantum systems can exist. This concept states that unlike classical mechanics, where energy can vary continuously, in quantum mechanics, energy levels are discrete and determined by specific quantum conditions.
The energy levels are given by formulas that include fundamental constants such as Planck's constant.One example is the energy of an electron in a box, which is given by \(E_n = \frac{n^2 h^2}{8mL^2} \).
This quantization is key in explaining phenomena like the quantum jumps of electrons in atoms.
Boltzmann constant
The Boltzmann constant, denoted as \( k_B \), is a fundamental physical constant that connects temperature to energy. It has a value of \( 1.38 \times 10^{-23} \text{J/K} \).
The Boltzmann constant plays a crucial role in statistical mechanics and thermodynamics.
In the context of temperature and energy, the average energy of particles within a gas can be related to the temperature by the formula \( E = \frac{3}{2} k_B T \).
This allows us to understand how temperature affects the kinetic energy of particles in a system.
temperature and energy relationship
The relationship between temperature and energy in a gas can be calculated using the kinetic theory of gases.The average kinetic energy of a gas molecule is directly proportional to the temperature, given by the equation \( E = \frac{3}{2} k_B T \).
This means that as the temperature increases, the average kinetic energy of the particles also increases.Equating this to other energy forms (like the quantized energy levels of electrons), allows us to find the temperature corresponding to a specific energy level.
In the given exercise, we found the temperature at which the average energy matches the lowest energy of an electron in a box using this relationship.

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