To find the energy of one mole of photons of radiation with a given frequency, we can follow these steps:
### Step 1: Use 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 in joules,
- \( h \) is Planck's constant (\( 6.62 \times 10^{-34} \, \text{J s} \)),
- \( \nu \) is the frequency in hertz (\( 5 \times 10^{14} \, \text{Hz} \)).
### Step 2: Substitute the values into the formula
Substituting the values into the formula:
\[ E = (6.62 \times 10^{-34} \, \text{J s}) \times (5 \times 10^{14} \, \text{Hz}) \]
### Step 3: Calculate the energy of one photon
Now, perform the multiplication:
\[ E = 6.62 \times 5 \times 10^{-34} \times 10^{14} \]
\[ E = 33.1 \times 10^{-20} \, \text{J} \]
\[ E = 3.31 \times 10^{-19} \, \text{J} \]
### Step 4: Convert the energy from joules to kilojoules
To convert joules to kilojoules, use the conversion factor \( 1 \, \text{kJ} = 1000 \, \text{J} \):
\[ E = \frac{3.31 \times 10^{-19} \, \text{J}}{1000} \]
\[ E = 3.31 \times 10^{-22} \, \text{kJ} \]
### Step 5: Calculate the energy for one mole of photons
Since one mole of photons corresponds to Avogadro's number (\( 6.022 \times 10^{23} \) photons), we can find the energy for one mole:
\[ E_{\text{mole}} = E \times N_A \]
\[ E_{\text{mole}} = (3.31 \times 10^{-22} \, \text{kJ}) \times (6.022 \times 10^{23}) \]
### Step 6: Perform the multiplication
Calculating the energy for one mole:
\[ E_{\text{mole}} = 3.31 \times 6.022 \times 10^{-22 + 23} \]
\[ E_{\text{mole}} = 19.93 \, \text{kJ} \]
### Step 7: Round the result
Rounding the result gives:
\[ E_{\text{mole}} \approx 199 \, \text{kJ} \]
### Final Answer
The energy of one mole of photons of radiation with a frequency of \( 5 \times 10^{14} \, \text{Hz} \) is approximately **199 kJ**.
---
