energy of one mol of photons whose frequency is `5xx10^(14)Hz` is approximately equal to :

Options

A. `199kJ mol^(-1)`
B. `174 kJ mol^(-1)`
C. `150 kJ mol^(-1)`
D. `300 kJ mol^(-1)`

Explanation

To find the energy of one mole of photons given the frequency, we can follow these steps:

### Step 1: Understand the formula for energy of a photon
The energy \( E \) of a single photon can be calculated using the formula:
\[ E = h \nu \]
where:
- \( E \) is the energy of one photon,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( \nu \) is the frequency of the photon.

### Step 2: Substitute the given frequency into the formula
Given the frequency \( \nu = 5 \times 10^{14} \, \text{Hz} \), we can substitute this value into the formula:
\[ E = (6.626 \times 10^{-34} \, \text{J s}) \times (5 \times 10^{14} \, \text{Hz}) \]

### Step 3: Calculate the energy of one photon
Now, performing the multiplication:
\[ E = 6.626 \times 5 \times 10^{-34 + 14} \]
Calculating \( 6.626 \times 5 \):
\[ 6.626 \times 5 = 33.13 \]
Thus,
\[ E = 33.13 \times 10^{-20} \, \text{J} = 3.313 \times 10^{-19} \, \text{J} \]

### Step 4: Calculate the energy of one mole of photons
To find the energy of one mole of photons, we multiply the energy of one photon by Avogadro's number (\( N_A = 6.022 \times 10^{23} \)):
\[ E_{\text{mole}} = E \times N_A = (3.313 \times 10^{-19} \, \text{J}) \times (6.022 \times 10^{23}) \]

### Step 5: Perform the multiplication
Calculating this gives:
\[ E_{\text{mole}} = 3.313 \times 6.022 \times 10^{-19 + 23} \]
Calculating \( 3.313 \times 6.022 \):
\[ 3.313 \times 6.022 \approx 19.94 \]
Thus,
\[ E_{\text{mole}} \approx 19.94 \times 10^{4} \, \text{J} = 199.4 \, \text{J} \]

### Step 6: Final result
The energy of one mole of photons is approximately:
\[ E_{\text{mole}} \approx 199.4 \, \text{J} \]

### Conclusion
The answer is approximately equal to 199.4 J, which corresponds to option 1 in the provided choices.

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