To determine the longest wavelength of a photon that can initiate the reaction between hydrogen (H₂) and bromine (Br₂) to form hydrogen bromide (HBr), we need to consider the energy required to break the Br-Br bond. This energy is given as the bond dissociation energy of Br₂, which is 192.86 kJ/mol. We can use this information to find the corresponding wavelength of light that can provide this energy.
Understanding Energy and Wavelength Relationship
The energy of a photon is related to its wavelength through the equation:
E = hc/λ
- E is the energy of the photon (in joules).
- h is Planck's constant, approximately 6.626 × 10-34 J·s.
- c is the speed of light, approximately 3.00 × 108 m/s.
- λ is the wavelength (in meters).
Converting Energy Units
First, we need to convert the bond dissociation energy from kJ/mol to joules per photon. Since 1 mole contains Avogadro's number of particles (approximately 6.022 × 1023), we can use the following conversion:
E (in J) = (192.86 kJ/mol) × (1000 J/kJ) / (6.022 × 1023 mol-1)
Calculating this gives:
E ≈ 3.20 × 10-19 J
Finding the Wavelength
Now that we have the energy in joules, we can rearrange the energy-wavelength equation to solve for wavelength:
λ = hc/E
Substituting the values:
λ = (6.626 × 10-34 J·s) × (3.00 × 108 m/s) / (3.20 × 10-19 J)
Calculating this gives:
λ ≈ 6.22 × 10-7 m
Converting to Nanometers
Since it is often more convenient to express wavelengths in nanometers (1 nm = 10-9 m), we convert:
λ ≈ 622 nm
Conclusion
The longest wavelength of a photon that can initiate the reaction between H₂ and Br₂, leading to the formation of HBr, is approximately 622 nanometers. This wavelength falls within the visible spectrum, specifically in the orange-red region, indicating that light in this range can effectively break the Br-Br bond and initiate the reaction.
