Question

1. a photon of light has an energy of 4.42 x 10^(-19) j. calculate its frequency.
2. calculate the energy of a photon of red light with a wavelength of 645 nm. (hint: convert nm to m)
3. calculate the energy of a photon of light with a frequency of 2.40 x 10^(12) hz. identify the type of radiation.

Explanation:

Step1: Recall the energy - frequency formula

The energy of a photon is given by $E = h
u$, where $E$ is the energy, $h = 6.63\times10^{-34}\text{ J}\cdot\text{s}$ is Planck's constant and $
u$ is the frequency. We can solve for $
u$: $
u=\frac{E}{h}$.

Step2: Calculate the frequency for the first problem

Given $E = 4.42\times 10^{-19}\text{ J}$, then $
u=\frac{4.42\times 10^{-19}\text{ J}}{6.63\times 10^{-34}\text{ J}\cdot\text{s}}\approx6.67\times 10^{14}\text{ Hz}$.

Step3: Recall the energy - wavelength formula

The energy of a photon can also be expressed as $E=\frac{hc}{\lambda}$, where $c = 3\times 10^{8}\text{ m/s}$ is the speed of light and $\lambda$ is the wavelength. First, convert $\lambda=645\text{ nm}=645\times 10^{-9}\text{ m}$.

Step4: Calculate the energy for the second problem

$E=\frac{6.63\times 10^{-34}\text{ J}\cdot\text{s}\times3\times 10^{8}\text{ m/s}}{645\times 10^{-9}\text{ m}}\approx3.08\times 10^{-19}\text{ J}$.

Step5: Calculate the energy for the third problem

Using $E = h
u$, with $
u = 2.40\times 10^{12}\text{ Hz}$ and $h = 6.63\times 10^{-34}\text{ J}\cdot\text{s}$, we get $E=6.63\times 10^{-34}\text{ J}\cdot\text{s}\times2.40\times 10^{12}\text{ Hz}\approx1.59\times 10^{-21}\text{ J}$. This frequency corresponds to infrared radiation (since frequencies in the range of $3\times 10^{11}-4\times 10^{14}\text{ Hz}$ are infrared).

Answer:

1. $6.67\times 10^{14}\text{ Hz}$
2. $3.08\times 10^{-19}\text{ J}$
3. $1.59\times 10^{-21}\text{ J}$, Infrared radiation