What is the band gap energy in kilojoules per mole if blue diamonds absorb orange light with a wavelength of 675 nm?

To find the band gap energy in kilojoules per mole for blue diamonds absorbing orange light with a wavelength of 675 nm, we can use the following relationship between the energy of a photon and its wavelength:

\[
E = \frac{h \cdot c}{\lambda}
\]

Where:
- \( E \) is the energy of the photon (in joules),
- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J} \cdot \text{s}) \),
- \( c \) is the speed of light \( (3.00 \times 10^8 \, \text{m/s}) \),
- \( \lambda \) is the wavelength of light (in meters).

Let's calculate the energy of one photon and then convert it to kilojoules per mole.

### Step 1: Convert the wavelength to meters
Since the given wavelength is 675 nm, we need to convert it to meters:
\[
675 \, \text{nm} = 675 \times 10^{-9} \, \text{m} = 6.75 \times 10^{-7} \, \text{m}
\]

### Step 2: Calculate the energy of one photon
\[
E_{\text{photon}} = \frac{6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \times 3.00 \times 10^8 \, \text{m/s}}{6.75 \times 10^{-7} \, \text{m}}
\]
\[
E_{\text{photon}} = 2.944 \times 10^{-19} \, \text{J}
\]

### Step 3: Convert the energy to kilojoules per mole
Now, we need to convert this energy for a mole of photons. One mole of photons contains Avogadro's number \( (6.022 \times 10^{23}) \) of photons. So the energy per mole of photons is:

\[
E_{\text{mole}} = E_{\text{photon}} \times 6.022 \times 10^{23}
\]
\[
E_{\text{mole}} = 2.944 \times 10^{-19} \, \text{J} \times 6.022 \times 10^{23}
\]
\[
E_{\text{mole}} = 1.772 \times 10^5 \, \text{J/mol} = 177.2 \, \text{kJ/mol}
\]

### Conclusion:
The band gap energy of blue diamonds absorbing orange light with a wavelength of 675 nm is approximately **177.2 kJ/mol**.

Do you need further details or have any questions?

Here are 5 related questions:
1. How does the band gap energy of a material affect its optical properties?
2. What is the relationship between the color of a material and the wavelength of light it absorbs?
3. Can we calculate the band gap energy for materials absorbing light of different wavelengths?
4. What is the significance of Avogadro's number in energy calculations involving moles?
5. How does the energy of a photon change with respect to its wavelength?

**Tip**: Shorter wavelengths correspond to higher energy photons, meaning materials absorbing blue or violet light have higher band gap energies than those absorbing red or orange light.

Calculate the band gap energy for blue diamonds absorbing orange light with a wavelength of 675 nm. This problem involves Planck's constant, the speed of light, and Avogadro's number to determine the energy in kilojoules per mole. Suitable for Grades 11-12 in advanced chemistry or physics.

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