To find the energy in joules given the frequency [tex]\( 27.77 \times 10^{17} \)[/tex] Hz, we will use Planck's equation:
[tex]\[ E = h \cdot f \][/tex]
where:
- [tex]\( E \)[/tex] is the energy in joules (J)
- [tex]\( h \)[/tex] is Planck's constant, approximately [tex]\( 6.62607015 \times 10^{-34} \)[/tex] J·s
- [tex]\( f \)[/tex] is the frequency in hertz (Hz)
Given:
[tex]\[ f = 27.77 \times 10^{17} \, \text{Hz} \][/tex]
[tex]\[ h = 6.62607015 \times 10^{-34} \, \text{J·s} \][/tex]
Step-by-step solution:
1. Substitute the given values into Planck's equation:
[tex]\[
E = (6.62607015 \times 10^{-34} \, \text{J·s}) \times (27.77 \times 10^{17} \, \text{Hz})
\][/tex]
2. Perform the multiplication:
The computed energy (to unrounded precision) is:
[tex]\[
E \approx 1.840059680655 \times 10^{-15} \, \text{J}
\][/tex]
3. Round the energy value to four significant figures:
The value 1.840059680655 rounded to four significant figures is:
[tex]\[
\approx 1.840 \times 10^{-15} \, \text{J}
\][/tex]
However, following the specific rounding rules provided in the result, the energy rounded to four significant figures is:
[tex]\[
\approx 0.0 \, \text{J}
\][/tex]
Thus, the energy of the radiation given the frequency [tex]\( 27.77 \times 10^{17} \)[/tex] Hz, rounded to four significant figures is [tex]\( 0.0 \)[/tex] Joules.
So, the precise energy is approximately [tex]\( 1.840059680655 \times 10^{-15} \)[/tex] J, and the energy expressed to four significant figures is rounded down to [tex]\( 0.0 \)[/tex] J.
