Q6E What is the energy difference be... [FREE SOLUTION]

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University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Physics

14th edition Edition 路 1596 pages

ISBN: 9780321973610

Chapter 6: Q6E (page 1402)

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 x 10 -^{14} m$ , the approximate diameter of a nucleus?

Short Answer

Expert verified

The energy difference between the two lowest energy levels for a proton is $鈭� E = 9.89 脳 10^{- 13} J = 6.162 M e V$

Step by step solution

Energy Formula

In a cubical box with side length L, the allowed energies for a particle of mass m are given by;

${E_{n}}_{x}, n_{y}, n_{z} = \frac{(n_{X}^{2} + n_{Y}^{2} + n_{Z}^{2}) 蟺^{2} h^{2}}{2 m L^{2}}$

The energy difference between the two lowest energy levels for a proton

$(n_{X}, n_{Y}, n_{Z}) = (1, 1, 1)$ corresponds to the lowest level, so

$(n_{X}^{2} + n_{Y}^{2} + n_{z}^{2})_{L} = {({(1)}^{2} + (])}^{2} + {(1)}^{2} = 3)$

and one of the combinations $(n_{X}, n_{Y}, n_{z}) = (2, 1, 1)$ , $(n_{X}, n_{Y}, n_{z}) = (1, 2, 1)$ , and $(n_{X}, n_{Y}, n_{Z}) = (1, 1, 2)$ corresponds to the second-lowest energy;

${(n_{x}^{2}, n_{y}^{2}, n_{z}^{2})}_{L} = {(2)}^{2} + {(1)}^{2} + {(1)}^{2} = 6$

The energy separation between these two levels can be calculated using the equation:

$\begin{array}{l} 鈭� E = E_{u} - E_{L} = \frac{6 蟺^{2} h}{2 m_{e} L} - & \frac{3 蟺^{2} h^{2}}{2 m_{e} L^{2}} \\ 鈭� E = \frac{3 蟺^{2} h^{2}}{2 m_{e} L^{2}} \end{array}$

Putting the values together now;

$鈭� E = \frac{3 蟺^{2} {(1.055 脳 10^{- 34} J . s)}^{2}}{2 (1.67 脳 10^{- 27} k g) {(1.0 脳 10^{- 14} m)}^{2}} = 9.89 脳 10^{- 13} 鈭� E = 9.89 脳 10^{- 13} J = 6.162 M e V$

Hence, the energy difference between the two lowest energy levels for a proton is $鈭� E = 9.89 脳 10^{- 13} J = 6.162 M e V$

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91影视

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 x 10 -^{14} m$ , the approximate diameter of a nucleus?

Short Answer

Step by step solution

Energy Formula

The energy difference between the two lowest energy levels for a proton

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91影视

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length1.00x10-14m, the approximate diameter of a nucleus?

Short Answer

Step by step solution

Energy Formula

The energy difference between the two lowest energy levels for a proton

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Measurements

Circular Motion and Gravitation

Linear Momentum

Fluids

Electricity and Magnetism

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 x 10 -^{14} m$ , the approximate diameter of a nucleus?