Calculate the wavelength (in nm) of a neutron that has a velocity of 200. cm/s. (The mass of a neutron = 1.675 10-24 g). Express your answer in decimal notation rounded to three significant figures.
Calculate the wavelength (in nm) of a neutron that has a velocity of 200. cm/s. (The mass of a neutron = 1.675 10-24 g). Express your answer in decimal notation rounded to three significant figures.
Chemistry: Matter and Change
1st Edition
ISBN:9780078746376
Author:Dinah Zike, Laurel Dingrando, Nicholas Hainen, Cheryl Wistrom
Publisher:Dinah Zike, Laurel Dingrando, Nicholas Hainen, Cheryl Wistrom
Chapter5: Electrons In The Atom
Section5.1: Light And Quantized Energy
Problem 7PP
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![### Problem Statement
Calculate the wavelength (in nm) of a neutron that has a velocity of 200. cm/s.
(The mass of a neutron = 1.675 × 10^-24 g).
Express your answer in decimal notation rounded to three significant figures.
### Explanation
This problem involves calculating the wavelength of a neutron using its velocity and mass. The formula employed to find the wavelength (λ) of a particle is derived from the de Broglie wavelength equation:
\[ \lambda = \frac{h}{mv} \]
where:
- \( \lambda \) is the wavelength,
- \( h \) is the Planck constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.
### Step-by-Step Solution
1. Convert the mass of the neutron from grams to kilograms:
\[ m = 1.675 \times 10^{-24} \, \text{g} \times \frac{1 \, \text{kg}}{10^3 \, \text{g}} = 1.675 \times 10^{-27} \, \text{kg} \]
2. Use the de Broglie equation to find the wavelength:
\[ \lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34} \, \text{Js}}{(1.675 \times 10^{-27} \, \text{kg})(200 \, \text{cm/s})}\]
**Note:** Convert the velocity from cm/s to m/s:
\[ 200 \, \text{cm/s} = 2 \, \text{m/s} \]
Substituting these values into the equation:
\[ \lambda = \frac{6.626 \times 10^{-34}}{(1.675 \times 10^{-27})(2)} \]
\[ \lambda = \frac{6.626 \times 10^{-34}}{3.35 \times 10^{-27}} \]
\[ \lambda \approx 1.977 \times 10^{-7} \, \text{m} \]
3. Convert the wavelength from meters to nanometers (1 m = \(10^9\) nm):
\[ \lambda \approx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd321eb40-4702-45a2-842a-571d0c67a0bc%2Fa397484c-6dba-44d5-933b-92c95c50fd94%2Ffs6zg8n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Calculate the wavelength (in nm) of a neutron that has a velocity of 200. cm/s.
(The mass of a neutron = 1.675 × 10^-24 g).
Express your answer in decimal notation rounded to three significant figures.
### Explanation
This problem involves calculating the wavelength of a neutron using its velocity and mass. The formula employed to find the wavelength (λ) of a particle is derived from the de Broglie wavelength equation:
\[ \lambda = \frac{h}{mv} \]
where:
- \( \lambda \) is the wavelength,
- \( h \) is the Planck constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.
### Step-by-Step Solution
1. Convert the mass of the neutron from grams to kilograms:
\[ m = 1.675 \times 10^{-24} \, \text{g} \times \frac{1 \, \text{kg}}{10^3 \, \text{g}} = 1.675 \times 10^{-27} \, \text{kg} \]
2. Use the de Broglie equation to find the wavelength:
\[ \lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34} \, \text{Js}}{(1.675 \times 10^{-27} \, \text{kg})(200 \, \text{cm/s})}\]
**Note:** Convert the velocity from cm/s to m/s:
\[ 200 \, \text{cm/s} = 2 \, \text{m/s} \]
Substituting these values into the equation:
\[ \lambda = \frac{6.626 \times 10^{-34}}{(1.675 \times 10^{-27})(2)} \]
\[ \lambda = \frac{6.626 \times 10^{-34}}{3.35 \times 10^{-27}} \]
\[ \lambda \approx 1.977 \times 10^{-7} \, \text{m} \]
3. Convert the wavelength from meters to nanometers (1 m = \(10^9\) nm):
\[ \lambda \approx
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