Answer
Frequency \( 3.00 \times 10^{10} \, \text{Hz} \)
Energy of a single photon: \( 2 \times 10^{-23} \, \text{J} \)
Work Step by Step
To calculate the frequency (\( f \)) of the microwave radiation, we can use the formula:
\[ f = \frac{c}{\lambda} \]
where \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)) and \( \lambda \) is the wavelength (\( 1.0 \, \text{cm} \)). First, we need to convert the wavelength to meters:
\[ \lambda = 1.0 \, \text{cm} \times 10^{-2} \, \text{m/cm} = 0.01 \, \text{m} \]
Now we can calculate the frequency:
\[ f = \frac{3.00 \times 10^8 \, \text{m/s}}{0.01 \, \text{m}} = 3.00 \times 10^{10} \, \text{Hz} \]
Now, to find the energy of a single photon, we can use Planck's equation:
\[ E = h \cdot f \] where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \)) and \( f \) is the frequency we just calculated. Let's calculate:
\[ E = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \times 3.00 \times 10^{10} \, \text{Hz} \] \[ E = 1.99 \times 10^{-23} \, \text{J} \] So, the frequency of the microwave radiation is \( 3.00 \times 10^{10} \, \text{Hz} \), and the energy of a single photon of this radiation is approximately \(2 \times 10^{-23} \, \text{J} \).
