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.