Calculate ther energy in kilocalorie per mol of the photons of an electromagnetic radiation of wavelength 5700Ã…

Answer

To calculate the energy in kilocalories per mole of photons of electromagnetic radiation with a wavelength of 5700 Å, follow these steps:

### Step 1: Convert Wavelength to Meters
The given wavelength is 5700 Å (angstroms). To convert this to meters:
\[ 1 \text{ Å} = 10^{-10} \text{ m} \]
So,
\[ 5700 \text{ Å} = 5700 \times 10^{-10} \text{ m} = 5.7 \times 10^{-7} \text{ m} \]

### Step 2: Calculate Frequency
Using the speed of light \( c = 3 \times 10^8 \text{ m/s} \), we can find the frequency \( \nu \) using the formula:
\[ \nu = \frac{c}{\lambda} \]
Substituting the values:
\[ \nu = \frac{3 \times 10^8 \text{ m/s}}{5.7 \times 10^{-7} \text{ m}} \approx 5.26 \times 10^{14} \text{ Hz} \]

### Step 3: Calculate Energy of a Photon
The energy \( E \) of a photon is given by:
\[ E = h \nu \]
Where \( h \) (Planck's constant) is approximately \( 6.626 \times 10^{-34} \text{ J s} \). Thus,
\[ E = 6.626 \times 10^{-34} \text{ J s} \times 5.26 \times 10^{14} \text{ Hz} \approx 3.49 \times 10^{-19} \text{ J} \]

### Step 4: Convert Energy to Kilocalories
To convert joules to kilocalories, we use the conversion factor:
\[ 1 \text{ kcal} = 4184 \text{ J} \]
Thus,
\[ E \text{ (in kcal)} = \frac{3.49 \times 10^{-19} \text{ J}}{4184 \text{ J/kcal}} \approx 8.35 \times 10^{-23} \text{ kcal} \]

### Step 5: Calculate Energy per Mole
To find the energy per mole, multiply the energy of one photon by Avogadro's number (\( N_A \approx 6.022 \times 10^{23} \text{ mol}^{-1} \)):
\[ E \text{ (per mole)} = 8.35 \times 10^{-23} \text{ kcal} \times 6.022 \times 10^{23} \text{ mol}^{-1} \approx 50.19 \text{ kcal/mol} \]

### Final Answer
The energy of the photons of electromagnetic radiation with a wavelength of 5700 Å is approximately **50.19 kilocalories per mole**.

---