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can you give me a similar problem to this one and solve it step by step please

Certainly! I'll create a similar problem and provide a detailed solution for it. Let's try this:

### Problem
1. Find the energy of:
   - (a) a photon having a frequency of \(6.50 \times 10^{15}\) Hz.
   - (b) a photon having a wavelength of \(4.00 \times 10^{2}\) nm.

Express your answers in units of electron volts (eV), noting that \(1 \, \text{eV} = 1.60 \times 10^{-19} \, \text{J}\).

### Solution

To solve this, we will use the following formulas:

1. **Energy from frequency**: \( E = h \cdot f \)
   - where \( E \) is the energy in joules, \( h \) is Planck's constant (\(6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\)), and \( f \) is the frequency.
2. **Energy from wavelength**: \( E = \frac{h \cdot c}{\lambda} \)
   - where \( c \) is the speed of light (\(3.00 \times 10^{8} \, \text{m/s}\)), and \( \lambda \) is the wavelength in meters.

Finally, to convert joules to electron volts, we use the conversion factor:
   \[
   1 \, \text{eV} = 1.60 \times 10^{-19} \, \text{J}
   \]

---

### Step-by-Step Solution

#### (a) Energy of a Photon with Frequency \(6.50 \times 10^{15}\) Hz

1. **Calculate Energy in Joules**:
   \[
   E = h \cdot f = (6.63 \times 10^{-34} \, \text{J} \cdot \text{s}) \cdot (6.50 \times 10^{15} \, \text{Hz})
   \]
   \[
   E = 4.3095 \times 10^{-18} \, \text{J}
   \]

2. **Convert to Electron Volts (eV)**:
   \[
   E = \frac{4.3095 \times 10^{-18} \, \text{J}}{1.60 \times 10^{-19} \, \text{J/eV}}
   \]
   \[
   E \approx 26.934 \, \text{eV}
   \]

So, the energy of a photon with a frequency of \(6.50 \times 10^{15}\) Hz is approximately **26.93 eV**.

---

#### (b) Energy of a Photon with Wavelength \(4.00 \times 10^{2}\) nm

1. **Convert Wavelength to Meters**:
   \[
   \lambda = 4.00 \times 10^{2} \, \text{nm} = 4.00 \times 10^{-7} \, \text{m}
   \]

2. **Calculate Energy in Joules**:
   \[
   E = \frac{h \cdot c}{\lambda} = \frac{(6.63 \times 10^{-34} \, \text{J} \cdot \text{s}) \cdot (3.00 \times 10^{8} \, \text{m/s})}{4.00 \times 10^{-7} \, \text{m}}
   \]
   \[
   E = 4.9725 \times 10^{-19} \, \text{J}
   \]

3. **Convert to Electron Volts (eV)**:
   \[
   E = \frac{4.9725 \times 10^{-19} \, \text{J}}{1.60 \times 10^{-19} \, \text{J/eV}}
   \]
   \[
   E \approx 3.108 \, \text{eV}
   \]

So, the energy of a photon with a wavelength of \(4.00 \times 10^{2}\) nm is approximately **3.11 eV**.

---

Would you like more details on any of the steps? 

### Related Questions
1. What would be the energy of a photon with a frequency of \(8.00 \times 10^{15}\) Hz?
2. How would the energy change if the wavelength were doubled?
3. Calculate the energy for a photon with a frequency of \(4.00 \times 10^{14}\) Hz in eV.
4. If the energy of a photon is 10 eV, what is its wavelength?
5. How would you express the energy of a photon in joules if given the frequency in MHz?

### Tip
Remember that as the wavelength decreases, the energy of the photon increases since they are inversely proportional.

Find the energy of (a) a photon having a frequency of 5.00 x 10^17 Hz and (b) a photon having a wavelength of 3.00 x 10^2 nm. Express your answers in units of electron volts, noting that 1 eV = 1.60 x 10^-19 J.

Learn how to calculate the energy of photons based on their frequency and wavelength, using Planck's equation and converting to electron volts (eV). This example problem involves calculating the energy for a photon with a frequency of 5.00 x 10^17 Hz and one with a wavelength of 3.00 x 10^2 nm, suitable for high school physics students.

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