- The energy required to break one H-I bond is \(\frac{299,000 \text{ J mol}^{-1}}{6.022 \times 10^{23} \text{ mol}^{-1}} \approx \boxed{4.965 \times 10^{-19} \text{ J}}\).
- The longest wavelength is \(\frac{hc}{E_{\text{bond}}} \approx \boxed{4.00 \times 10^{-7} \text{ m}}\).
- Since the energy of the photon at \(
u = 7.7 \times 10^{14} \text{ s}^{-1}\) is \(5.10 \times 10^{-19} \text{ J}\), which is greater than the bond energy, the photon \(\boxed{\text{has enough energy}}\).
Explanation
- Calculate the energy required to break one H-I bond
To break the H-I bond in a molecule of HI(g), we first determine the energy required to break a single bond. The given bond energy is \(299 \text{ kJ mol}^{-1}\). Since we need the energy in joules for a single bond, we perform the following conversion:
- Convert the bond energy from kJ to J:
\[299 \text{ kJ mol}^{-1} = 299 \times 10^3 \text{ J mol}^{-1} = 299,000 \text{ J mol}^{-1}\]
- Divide by Avogadro's number (\(N_A = 6.022 \times 10^{23} \text{ mol}^{-1}\)) to find the energy per bond (\(E_{\text{bond}}\)):
\[E_{\text{bond}} = \frac{299,000 \text{ J mol}^{-1}}{6.022 \times 10^{23} \text{ mol}^{-1}} \approx 4.965 \times 10^{-19} \text{ J}\]
Thus, the minimum energy required to break one H-I bond is approximately \(4.965 \times 10^{-19} \text{ J}\).
- Calculate the longest wavelength of light required
To find the longest wavelength of light (\(\lambda\)) that can supply this energy, we use the relationship between the energy of a photon and its wavelength: \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\)) and \(c\) is the speed of light (\(3.00 \times 10^8 \text{ m/s}\)).
Rearranging the equation to solve for \(\lambda\):
\[\lambda = \frac{hc}{E_{\text{bond}}}\]
\[\lambda = \frac{(6.626 \times 10^{-34} \text{ J s}) \times (3.00 \times 10^8 \text{ m/s})}{4.965 \times 10^{-19} \text{ J}} \approx 4.00 \times 10^{-7} \text{ m}\]
The longest wavelength capable of breaking the H-I bond is approximately \(4.00 \times 10^{-7} \text{ m}\).
- Determine if the photon energy is sufficient
To determine if a photon with a frequency (\(
u\)) of \(7.7 \times 10^{14} \text{ s}^{-1}\) has enough energy, we calculate its energy using the relation \(E_{\text{photon}} = h
u\):
\[E_{\text{photon}} = (6.626 \times 10^{-34} \text{ J s}) \times (7.7 \times 10^{14} \text{ s}^{-1}) \approx 5.10 \times 10^{-19} \text{ J}\]
Comparing this to the energy required to break the H-I bond (\(4.965 \times 10^{-19} \text{ J}\)):
Since \(5.10 \times 10^{-19} \text{ J} > 4.965 \times 10^{-19} \text{ J}\), the energy of the photon is sufficient to break the H-I bond. Thus, the answer is yes.
Examples
In chemistry, understanding the energy of photons is essential for photochemistry, which includes processes like photosynthesis and the degradation of atmospheric pollutants. For instance, determining whether UV light has enough energy to break specific bonds in greenhouse gases helps scientists predict how these compounds decompose in the atmosphere, influencing our climate models. By applying these calculations, researchers can determine the threshold wavelengths required to trigger specific chemical reactions, providing a quantitative basis for studying the interactions between light and matter.