What is the energy of a photon with a frequency of [tex]$1.7 \times 10^{17} Hz$[/tex]? Planck's constant is [tex]$6.63 \times 10^{-34} J \cdot s$[/tex].

A. [tex][tex]$1.1 \times 10^{-173} J$[/tex][/tex]
B. [tex]$1.1 \times 10^{-16} J$[/tex]
C. [tex]$8.3 \times 10^{-16} J$[/tex]
D. [tex][tex]$8.3 \times 10^{-15} J$[/tex][/tex]



Answer :

Let's solve the problem step-by-step:

1. Identify the given quantities:
- The frequency of the photon ([tex]\( \nu \)[/tex]) is [tex]\( 1.7 \times 10^{17} \, \text{Hz} \)[/tex].
- Planck's constant ([tex]\( h \)[/tex]) is [tex]\( 6.63 \times 10^{-34} \, \text{J} \cdot \text{s} \)[/tex].

2. Recall the formula to calculate the energy of a photon:
[tex]\[ E = h \nu \][/tex]
where
[tex]\( E \)[/tex] is the energy,
[tex]\( h \)[/tex] is Planck's constant, and
[tex]\( \nu \)[/tex] is the frequency.

3. Substitute the given values into the formula:
[tex]\[ E = (6.63 \times 10^{-34} \, \text{J} \cdot \text{s}) \times (1.7 \times 10^{17} \, \text{Hz}) \][/tex]

4. Perform the multiplication:
[tex]\[ E = 6.63 \times 1.7 \times 10^{-34 + 17} \, \text{J} \][/tex]

5. Simplify the numerical part:
[tex]\[ 6.63 \times 1.7 = 11.271 \][/tex]

And combine the exponents:
[tex]\[ 10^{-34 + 17} = 10^{-17} \][/tex]

6. Combine the numerical and exponential parts:
[tex]\[ E = 11.271 \times 10^{-17} \, \text{J} \][/tex]

7. Convert to scientific notation:
[tex]\[ E \approx 1.1271 \times 10^{-16} \, \text{J} \][/tex]

8. Round appropriately and match to the given options:
The closest option is:
[tex]\[ 1.1 \times 10^{-16} \, \text{J} \][/tex]

Therefore, the energy of the photon is [tex]\( 1.1 \times 10^{-16} \, \text{J} \)[/tex].