Sure, let's tackle these problems step-by-step with all the necessary equations and substitutions.
### Part a: Calculate the wavelength of the photon
Given:
- Energy of the photon ([tex]\( E \)[/tex]) = [tex]\( 3.26 \times 10^{-19} \)[/tex] joules
- Speed of light ([tex]\( c \)[/tex]) = [tex]\( 3 \times 10^8 \)[/tex] meters per second
- Planck's constant ([tex]\( h \)[/tex]) = [tex]\( 6.626 \times 10^{-34} \)[/tex] joule seconds
Equation:
To find the wavelength ([tex]\( \lambda \)[/tex]) of the photon, we use the energy-wavelength relationship given by the equation:
[tex]\[ E = \frac{h \cdot c}{\lambda} \][/tex]
Rearrange the equation to solve for [tex]\( \lambda \)[/tex]:
[tex]\[ \lambda = \frac{h \cdot c}{E} \][/tex]
Substitution:
Now, plug in the given values:
[tex]\[ \lambda = \frac{(6.626 \times 10^{-34} \text{ Js}) \times (3 \times 10^8 \text{ m/s})}{3.26 \times 10^{-19} \text{ J}} \][/tex]
Calculation:
[tex]\[ \lambda = \frac{1.9878 \times 10^{-25} \text{ Js m/s}}{3.26 \times 10^{-19} \text{ J}} \][/tex]
Perform the division:
[tex]\[ \lambda \approx 6.097546012269938 \times 10^{-7} \text{ meters} \][/tex]
So, the wavelength of the photon is approximately [tex]\( 6.097546012269938 \times 10^{-7} \)[/tex] meters.
### Part b: Calculate the photon's frequency
Given:
- Energy of the photon ([tex]\( E \)[/tex]) = [tex]\( 3.26 \times 10^{-19} \)[/tex] joules
- Planck's constant ([tex]\( h \)[/tex]) = [tex]\( 6.626 \times 10^{-34} \)[/tex] joule seconds
Equation:
To find the frequency ([tex]\( \nu \)[/tex]) of the photon, we use the energy-frequency relationship given by the equation:
[tex]\[ E = h \cdot \nu \][/tex]
Rearrange the equation to solve for [tex]\( \nu \)[/tex]:
[tex]\[ \nu = \frac{E}{h} \][/tex]
Substitution:
Now, plug in the given values:
[tex]\[ \nu = \frac{3.26 \times 10^{-19} \text{ J}}{6.626 \times 10^{-34} \text{ J s}} \][/tex]
Calculation:
[tex]\[ \nu \approx 492001207364926.06 \text{ Hz} \][/tex]
So, the frequency of the photon is approximately [tex]\( 4.920 \times 10^{14} \)[/tex] Hz, or more precisely, [tex]\( 492001207364926.06 \)[/tex] Hz.
This is the detailed step-by-step solution to calculate both the wavelength and the frequency of the photon given its energy.
