Calculate the threshold frequency,  in kJ/mol of electrons in aluminum, given that the lowest frequency photon for which the photoelectric effect is observed is 9.87 x 1014 Hz. Report answer with correct significant figures.

The threshold frequency \( f_0 \) is the minimum frequency of light required to eject electrons from a material in the photoelectric effect. Given the frequency \( f = 9.87 \times 10^{14} \, \text{Hz} \) for aluminum, we can calculate the energy per electron in joules using the equation for the energy of a photon:

\[
E = h f
\]

where:
- \( h = 6.626 \times 10^{-34} \, \text{J·s} \) is Planck's constant,
- \( f = 9.87 \times 10^{14} \, \text{Hz} \) is the frequency of the photon.

### Step 1: Calculate the energy per photon

\[
E = (6.626 \times 10^{-34} \, \text{J·s}) \times (9.87 \times 10^{14} \, \text{Hz}) = 6.537 \times 10^{-19} \, \text{J}
\]

### Step 2: Convert energy to kJ/mol

To convert the energy per photon to kJ/mol, we multiply by Avogadro's number \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \) and convert joules to kilojoules:

\[
E_{\text{mol}} = \left( 6.537 \times 10^{-19} \, \text{J} \right) \times (6.022 \times 10^{23} \, \text{mol}^{-1}) = 3.938 \times 10^{5} \, \text{J/mol}
\]

Now, convert to kJ/mol:

\[
E_{\text{mol}} = 393.8 \, \text{kJ/mol}
\]

### Final Answer:
The threshold frequency energy for aluminum is **393.8 kJ/mol**.

Let me know if you'd like more details on any of the steps.

Here are five related questions to expand understanding:
1. How does the threshold frequency relate to the work function in the photoelectric effect?
2. How would the result change if the frequency were lower than the threshold frequency?
3. What is the significance of Planck’s constant in calculating photon energy?
4. Can the threshold frequency vary between different materials, and if so, why?
5. How does Einstein's photoelectric equation relate to the energy calculated?

**Tip**: Always check the units in your calculations, especially when converting between J and kJ, or when applying constants like Avogadro's number.

Calculate the threshold frequency, in kJ/mol of electrons in aluminum, given that the lowest frequency photon for which the photoelectric effect is observed is 9.87 x 10^14 Hz. Report answer with correct significant figures.

Learn how to calculate the threshold frequency in kJ/mol of electrons in aluminum using photon energy. This problem involves the photoelectric effect, Planck's constant, and Avogadro's number. Detailed steps and conversion from joules to kilojoules per mole included. Suitable for advanced physics students in Grades 11-12.

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