To find the energy of a photon given its frequency, we can use the well-known formula from quantum mechanics:
[tex]\[ E = h \cdot f \][/tex]
where:
- [tex]\( E \)[/tex] is the energy of the photon,
- [tex]\( h \)[/tex] is Planck's constant ([tex]\( 6.63 \times 10^{-34} \)[/tex] J·s), and
- [tex]\( f \)[/tex] is the frequency of the photon ([tex]\( 2.53 \times 10^{12} \)[/tex] Hz).
Let's go through the steps to calculate the energy:
1. Identify the given values:
- Frequency ([tex]\( f \)[/tex]): [tex]\( 2.53 \times 10^{12} \)[/tex] Hz
- Planck's constant ([tex]\( h \)[/tex]): [tex]\( 6.63 \times 10^{-34} \)[/tex] J·s
2. Substitute the values into the formula:
[tex]\[
E = (6.63 \times 10^{-34} \text{ J·s}) \times (2.53 \times 10^{12} \text{ Hz})
\][/tex]
3. Multiply the numbers and the exponents:
[tex]\[
E = 6.63 \times 2.53 \times 10^{-34 + 12} \text{ J}
\][/tex]
4. Calculate the product of the coefficients:
[tex]\[
6.63 \times 2.53 = 16.7679
\][/tex]
5. Combine the exponents:
[tex]\[
10^{-34 + 12} = 10^{-22}
\][/tex]
6. Put it all together:
[tex]\[
E = 16.7679 \times 10^{-22} \text{ J}
\][/tex]
7. Adjust the number to proper scientific notation:
[tex]\[
E \approx 1.67739 \times 10^{-21} \text{ J}
\][/tex]
Now that we have calculated the energy to be roughly [tex]\( 1.67739 \times 10^{-21} \)[/tex] J, we need to compare this result with the given choices to find the closest match.
Here are the choices provided:
- [tex]\( 1.68 \times 10^{23} \)[/tex] J
- [tex]\( 1.68 \times 10^{47} \)[/tex] J
- [tex]\( 1.68 \times 10^{-21} \)[/tex] J
- [tex]\( 1.68 \times 10^{-45} \)[/tex] J
The closest and most accurate match is [tex]\( 1.68 \times 10^{-21} \)[/tex] J. Thus, the correct answer is:
[tex]\[ \boxed{1.68 \times 10^{-21} \text{ J}}. \][/tex]
