Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To solve this problem, we need to determine the wavelength of a quantum of light given certain parameters. We will use the relationship between the speed of light, frequency, and wavelength.
Here's the step-by-step solution:
1. Given Values:
- Planck's constant, [tex]\( h \)[/tex] = [tex]\( 6.63 \times 10^{-34} \)[/tex] Js (though this value isn't directly needed for calculating wavelength, it's given in the problem).
- Speed of light, [tex]\( c \)[/tex] = [tex]\( 3.0 \times 10^8 \)[/tex] m/s.
- Frequency, [tex]\( f \)[/tex] = [tex]\( 8 \times 10^{15} \)[/tex] Hz.
2. Formula for Wavelength:
The wavelength ([tex]\( \lambda \)[/tex]) of light can be calculated using the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\( \lambda \)[/tex] is the wavelength,
- [tex]\( c \)[/tex] is the speed of light,
- [tex]\( f \)[/tex] is the frequency.
3. Calculate the Wavelength in Meters:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \text{ m/s}}{8 \times 10^{15} \text{ Hz}} \][/tex]
[tex]\[ \lambda = \frac{3.0}{8} \times 10^{8-15} \text{ m} \][/tex]
[tex]\[ \lambda = 0.375 \times 10^{-7} \text{ m} \][/tex]
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \][/tex]
4. Convert the Wavelength to Nanometers:
Since 1 meter = [tex]\( 1 \times 10^9 \)[/tex] nanometers:
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \times 10^9 \text{ nm/m} \][/tex]
[tex]\[ \lambda = 37.5 \text{ nm} \][/tex]
5. Select the Closest Value from the Given Options:
The wavelengths provided in the choices are in different scales. Let's review them and see which one is closest to our calculated value of 37.5 nm.
- Option (a): [tex]\( 3 \times 10^7 \)[/tex] nm (this is [tex]\( 30,000,000 \)[/tex] nm - much too large).
- Option (b): [tex]\( 2 \times 10^{-25} \)[/tex] nm (this is [tex]\( 0.000...00002 \)[/tex] nm - much too small).
- Option (c): [tex]\( 5 \times 10^{-18} \)[/tex] nm (this is [tex]\( 0.000...00005 \)[/tex] nm - also too small).
- Option (d): [tex]\( 4 \times 10^1 \)[/tex] nm (this is [tex]\( 40 \)[/tex] nm - close to 37.5 nm).
Therefore, the value closest to the wavelength in nanometers of a quantum of light with the given frequency is:
[tex]\[ \boxed{4 \times 10^1} \][/tex]
So, the correct option is (d) [tex]\( 4 \times 10^1 \)[/tex] nm.
Here's the step-by-step solution:
1. Given Values:
- Planck's constant, [tex]\( h \)[/tex] = [tex]\( 6.63 \times 10^{-34} \)[/tex] Js (though this value isn't directly needed for calculating wavelength, it's given in the problem).
- Speed of light, [tex]\( c \)[/tex] = [tex]\( 3.0 \times 10^8 \)[/tex] m/s.
- Frequency, [tex]\( f \)[/tex] = [tex]\( 8 \times 10^{15} \)[/tex] Hz.
2. Formula for Wavelength:
The wavelength ([tex]\( \lambda \)[/tex]) of light can be calculated using the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\( \lambda \)[/tex] is the wavelength,
- [tex]\( c \)[/tex] is the speed of light,
- [tex]\( f \)[/tex] is the frequency.
3. Calculate the Wavelength in Meters:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \text{ m/s}}{8 \times 10^{15} \text{ Hz}} \][/tex]
[tex]\[ \lambda = \frac{3.0}{8} \times 10^{8-15} \text{ m} \][/tex]
[tex]\[ \lambda = 0.375 \times 10^{-7} \text{ m} \][/tex]
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \][/tex]
4. Convert the Wavelength to Nanometers:
Since 1 meter = [tex]\( 1 \times 10^9 \)[/tex] nanometers:
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \times 10^9 \text{ nm/m} \][/tex]
[tex]\[ \lambda = 37.5 \text{ nm} \][/tex]
5. Select the Closest Value from the Given Options:
The wavelengths provided in the choices are in different scales. Let's review them and see which one is closest to our calculated value of 37.5 nm.
- Option (a): [tex]\( 3 \times 10^7 \)[/tex] nm (this is [tex]\( 30,000,000 \)[/tex] nm - much too large).
- Option (b): [tex]\( 2 \times 10^{-25} \)[/tex] nm (this is [tex]\( 0.000...00002 \)[/tex] nm - much too small).
- Option (c): [tex]\( 5 \times 10^{-18} \)[/tex] nm (this is [tex]\( 0.000...00005 \)[/tex] nm - also too small).
- Option (d): [tex]\( 4 \times 10^1 \)[/tex] nm (this is [tex]\( 40 \)[/tex] nm - close to 37.5 nm).
Therefore, the value closest to the wavelength in nanometers of a quantum of light with the given frequency is:
[tex]\[ \boxed{4 \times 10^1} \][/tex]
So, the correct option is (d) [tex]\( 4 \times 10^1 \)[/tex] nm.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.