Get the answers you've been searching for with IDNLearn.com. Our platform provides detailed and accurate responses from experts, helping you navigate any topic with confidence.
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 value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
