What is the energy associated with photons of wavelength 4000 A?

Options

A. `4.97xx10^(-19)J`
B. `4.97xx10^(-18)J`
C. `4.97xx10^(-19)erg`
D. none of these

Answer

To find the energy associated with photons of wavelength 4000 Å (angstroms), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between energy and frequency**:
The energy \( E \) of a photon is given by the equation:
\[ E = h \nu \]
where \( h \) is Planck's constant and \( \nu \) is the frequency of the photon.

2. **Relate frequency to wavelength**:
The frequency \( \nu \) can be expressed in terms of the speed of light \( c \) and the wavelength \( \lambda \):
\[ \nu = \frac{c}{\lambda} \]
Therefore, we can substitute this into the energy equation:
\[ E = h \frac{c}{\lambda} \]

3. **Identify the constants**:
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- Speed of light \( c = 3.00 \times 10^8 \, \text{m/s} \)
- Wavelength \( \lambda = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} = 4.0 \times 10^{-7} \, \text{m} \)

4. **Substitute the values into the energy equation**:
\[ E = h \frac{c}{\lambda} = (6.626 \times 10^{-34} \, \text{J s}) \cdot \left(\frac{3.00 \times 10^8 \, \text{m/s}}{4.0 \times 10^{-7} \, \text{m}}\right) \]

5. **Calculate the energy**:
First, calculate the fraction:
\[ \frac{3.00 \times 10^8}{4.0 \times 10^{-7}} = 7.5 \times 10^{14} \, \text{s}^{-1} \]
Now, substitute this back into the energy equation:
\[ E = 6.626 \times 10^{-34} \cdot 7.5 \times 10^{14} \]
\[ E = 4.9695 \times 10^{-19} \, \text{J} \]

6. **Round off the answer**:
The energy associated with photons of wavelength 4000 Å is approximately:
\[ E \approx 4.97 \times 10^{-19} \, \text{J} \]

### Final Answer:
The energy associated with photons of wavelength 4000 Å is \( 4.97 \times 10^{-19} \, \text{J} \).