To solve the problem, follow these steps:
1. Understand the Given Values:
- Energy required to heat the water (\(E_{\text{required}}\)): \(7.06 \times 10^4 \text{ J}\)
- Microwave frequency (\(\nu\)): \(2.88 \times 10^{10} \text{ Hz}\)
2. Use Planck’s Constant:
- Planck's constant (\(h\)): \(6.62607015 \times 10^{-34} \text{ J} \cdot \text{s}\)
3. Calculate the Energy of One Quantum (Photon):
- The energy (\(E_{\text{quantum}}\)) of one photon can be calculated using the formula:
[tex]\[
E_{\text{quantum}} = h \cdot \nu
\][/tex]
- Plug in the values:
[tex]\[
E_{\text{quantum}} = (6.62607015 \times 10^{-34} \text{ J} \cdot \text{s}) \cdot (2.88 \times 10^{10} \text{ Hz})
\][/tex]
- Calculate:
[tex]\[
E_{\text{quantum}} = 1.9083082032 \times 10^{-23} \text{ J}
\][/tex]
4. Calculate the Number of Quanta Required:
- The number of quanta (\(N\)) needed to supply the required energy is given by:
[tex]\[
N = \frac{E_{\text{required}}}{E_{\text{quantum}}}
\][/tex]
- Plug in the values:
[tex]\[
N = \frac{7.06 \times 10^4 \text{ J}}{1.9083082032 \times 10^{-23} \text{ J}}
\][/tex]
- Calculate:
[tex]\[
N = 3.6996120375949974 \times 10^{27}
\][/tex]
Therefore, the number of quanta required to supply the [tex]\(7.06 \times 10^4 \text{ J}\)[/tex] of energy is approximately [tex]\(3.70 \times 10^{27}\)[/tex].
