To express the energy in joules to three significant figures, we will use the known relationship between energy, frequency, and Planck's constant. The formula to calculate the energy ([tex]\(E\)[/tex]) is given by:
[tex]\[ E = h \times \nu \][/tex]
where:
- [tex]\(h\)[/tex] is Planck's constant.
- [tex]\(\nu\)[/tex] (nu) is the frequency of the radiation.
Given:
- The frequency [tex]\(\nu = 3.43 \times 10^{14}\)[/tex] Hz.
- Planck's constant [tex]\(h = 6.62607015 \times 10^{-34}\)[/tex] J·s (joule-seconds).
Now, calculate the energy [tex]\(E\)[/tex]:
[tex]\[ E = (6.62607015 \times 10^{-34} \, \text{J·s}) \times (3.43 \times 10^{14} \, \text{Hz}) \][/tex]
First, multiply the numerical parts:
[tex]\[ 6.62607015 \times 3.43 = 22.7274206145 \][/tex]
Then, combine the exponents of ten:
[tex]\[ 10^{-34} \times 10^{14} = 10^{-20} \][/tex]
So,
[tex]\[ E = 22.7274206145 \times 10^{-20} \][/tex]
Express this number in scientific notation:
[tex]\[ E = 2.27274206145 \times 10^{-19} \, \text{Joules} \][/tex]
To round this energy value to three significant figures, we look at the first three significant digits:
[tex]\[ E \approx 2.27 \times 10^{-19} \, \text{Joules} \][/tex]
Thus, the energy in joules to three significant figures is:
[tex]\[ 2.27 \times 10^{-19} \, \text{Joules} \][/tex]
