Sodium street lamp gives off a characteristic yellow light of wavelength 588 nm. Calculate the energy mole (in kJ/mol) of these photons.

Explanation

To calculate the energy per mole of photons emitted by a sodium street lamp with a wavelength of 588 nm, we will follow these steps:

### Step 1: Identify the given values
- Wavelength (λ) = 588 nm = 588 × 10^-9 m
- Avogadro's number (N₀) = 6.022 × 10²³ mol⁻¹
- Planck's constant (h) = 6.626 × 10^-34 J·s
- Speed of light (c) = 3.00 × 10^8 m/s

### Step 2: Calculate the frequency (ν) of the light
The frequency can be calculated using the formula:
\[ \nu = \frac{c}{\lambda} \]
Substituting the values:
\[ \nu = \frac{3.00 \times 10^8 \text{ m/s}}{588 \times 10^{-9} \text{ m}} \]
\[ \nu = 5.1 \times 10^{14} \text{ Hz} \]

### Step 3: Calculate the energy of a single photon
The energy (E) of a single photon can be calculated using the formula:
\[ E = h \nu \]
Substituting the values:
\[ E = (6.626 \times 10^{-34} \text{ J·s})(5.1 \times 10^{14} \text{ Hz}) \]
\[ E = 3.38 \times 10^{-19} \text{ J} \]

### Step 4: Calculate the energy per mole of photons
To find the energy per mole, we multiply the energy of a single photon by Avogadro's number:
\[ E_{\text{mole}} = N₀ \times E \]
Substituting the values:
\[ E_{\text{mole}} = (6.022 \times 10^{23} \text{ mol}^{-1})(3.38 \times 10^{-19} \text{ J}) \]
\[ E_{\text{mole}} = 2.03 \times 10^{5} \text{ J/mol} \]

### Step 5: Convert the energy from joules to kilojoules
To convert joules to kilojoules, divide by 1000:
\[ E_{\text{mole}} = \frac{2.03 \times 10^{5} \text{ J/mol}}{1000} \]
\[ E_{\text{mole}} = 203 \text{ kJ/mol} \]

### Final Answer
The energy per mole of photons emitted by the sodium street lamp is **203 kJ/mol**.

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