Ch 41: Atomic Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 41: Atomic Physics

Problem 40

Chapter 41, Problem 40

A sodium atom emits a photon with wavelength 818 nm shortly after being struck by an electron. What minimum speed did the electron have before the collision?

Verified step by step guidance

Step 1: Begin by understanding the relationship between the energy of the emitted photon and its wavelength. Use the formula for photon energy:

E = \frac{hc}{λ}

, where

h

is Planck's constant (

6.63 \times 10 ⁻ 34

J·s),

c

is the speed of light (

3.00 \times 10 ⁸

m/s), and

λ

is the wavelength of the photon (818 nm, converted to meters as

818 \times 10 ⁻ 9

m).

Step 2: Calculate the energy of the photon using the formula from Step 1. This energy represents the minimum energy transferred to the sodium atom during the collision.

Step 3: Recognize that the electron must have had at least this amount of kinetic energy before the collision to transfer it to the sodium atom. Use the kinetic energy formula for the electron:

K = \frac{1}{2} m v^{2}

, where

m

is the mass of the electron (

9.11 \times 10 ⁻ 31

kg) and

v

is its speed.

Step 4: Rearrange the kinetic energy formula to solve for the speed of the electron:

v = \sqrt{\frac{2}{K} m}

. Substitute the energy of the photon (calculated in Step 2) for

K

and the mass of the electron for

m

Step 5: Perform the substitution and simplify the expression to find the minimum speed of the electron. This speed corresponds to the energy required to emit the photon with a wavelength of 818 nm.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Emission

Photon emission occurs when an atom transitions from a higher energy state to a lower energy state, releasing energy in the form of light. The wavelength of the emitted photon is inversely related to the energy difference between these states, as described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength.

Energy Conservation

In physics, the principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the kinetic energy of the incoming electron is converted into the energy of the emitted photon, allowing us to relate the electron's speed to the energy of the photon produced.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. To find the minimum speed of the electron, we can equate its kinetic energy to the energy of the emitted photon, allowing us to solve for the electron's initial speed before the collision.

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