Which has a higher energy, a photon of violet light with wavelength 4000 A ˚ or a photon of red light with wavelength 7000 A ˚ ?

Answer

To determine which photon has a higher energy, we can use the formula for the energy of a photon:

\[ E = \frac{hc}{\lambda} \]

Where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Joule seconds),
- \( c \) is the speed of light (\( 3.0 \times 10^{8} \) meters per second),
- \( \lambda \) is the wavelength of the light in meters.

### Step 1: Convert the wavelengths from angstroms to meters
1 angstrom (A) = \( 1 \times 10^{-10} \) meters.

- For violet light:
\[ \lambda_{violet} = 4000 \, \text{A} = 4000 \times 10^{-10} \, \text{m} = 4.0 \times 10^{-7} \, \text{m} \]

- For red light:
\[ \lambda_{red} = 7000 \, \text{A} = 7000 \times 10^{-10} \, \text{m} = 7.0 \times 10^{-7} \, \text{m} \]

### Step 2: Calculate the energy of the violet light photon
Using the formula:
\[ E_{violet} = \frac{hc}{\lambda_{violet}} \]

Substituting the values:
\[ E_{violet} = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.0 \times 10^{8} \, \text{m/s})}{4.0 \times 10^{-7} \, \text{m}} \]

Calculating:
\[ E_{violet} = \frac{1.9878 \times 10^{-25} \, \text{J m}}{4.0 \times 10^{-7} \, \text{m}} \]
\[ E_{violet} = 4.9695 \times 10^{-19} \, \text{J} \approx 4.96 \times 10^{-19} \, \text{J} \]

### Step 3: Calculate the energy of the red light photon
Using the formula:
\[ E_{red} = \frac{hc}{\lambda_{red}} \]

Substituting the values:
\[ E_{red} = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.0 \times 10^{8} \, \text{m/s})}{7.0 \times 10^{-7} \, \text{m}} \]

Calculating:
\[ E_{red} = \frac{1.9878 \times 10^{-25} \, \text{J m}}{7.0 \times 10^{-7} \, \text{m}} \]
\[ E_{red} = 2.8397 \times 10^{-19} \, \text{J} \approx 2.83 \times 10^{-19} \, \text{J} \]

### Step 4: Compare the energies
- Energy of violet light photon: \( E_{violet} \approx 4.96 \times 10^{-19} \, \text{J} \)
- Energy of red light photon: \( E_{red} \approx 2.83 \times 10^{-19} \, \text{J} \)

### Conclusion
The photon of violet light has a higher energy than the photon of red light.