Question

problem 1: a photon of light has a frequency of 7.50×10¹⁴ hz. calculate the energy of this photon in joules.
problem 2: what is the frequency of a photon that has an energy of 1.99×10⁻²⁰ j?
problem 3: a radio wave has a wavelength of 3.0 meters. what is its frequency in hertz?
problem 4: a photon of blue light has a frequency of 6.50×10¹⁴ hz. what is the wavelength of this light in meters?
problem 5: a microwave has a wavelength of 0.012 meters. what is the energy of a single photon of this microwave radiation in joules?

Explanation:

Step1: Recall Planck's - equation and wave - speed formula

The energy of a photon is given by $E = hf$, where $h=6.63\times 10^{-34}\ J\cdot s$ is Planck's constant and $f$ is the frequency. The speed of light $c = 3.0\times 10^{8}\ m/s$, and $c=\lambda f$, where $\lambda$ is the wavelength and $f$ is the frequency.

Problem 1:

Step1: Identify the formula and values

We use $E = hf$. Given $f = 7.50\times 10^{14}\ Hz$ and $h = 6.63\times 10^{-34}\ J\cdot s$.
$E=(6.63\times 10^{-34}\ J\cdot s)\times(7.50\times 10^{14}\ Hz)$

Step2: Calculate the energy

$E = 6.63\times7.50\times 10^{-34 + 14}\ J=4.9725\times 10^{-19}\ J$

Problem 2:

Step1: Rearrange the energy formula

From $E = hf$, we can solve for $f$: $f=\frac{E}{h}$. Given $E = 1.99\times 10^{-20}\ J$ and $h = 6.63\times 10^{-34}\ J\cdot s$.
$f=\frac{1.99\times 10^{-20}\ J}{6.63\times 10^{-34}\ J\cdot s}$

Step2: Calculate the frequency

$f=\frac{1.99}{6.63}\times10^{-20 + 34}\ Hz\approx 3.00\times 10^{13}\ Hz$

Problem 3:

Step1: Use the wave - speed formula

From $c=\lambda f$, we can solve for $f$: $f=\frac{c}{\lambda}$. Given $c = 3.0\times 10^{8}\ m/s$ and $\lambda = 3.0\ m$.
$f=\frac{3.0\times 10^{8}\ m/s}{3.0\ m}$

Step2: Calculate the frequency

$f = 1.0\times 10^{8}\ Hz$

Problem 4:

Step1: Rearrange the wave - speed formula

From $c=\lambda f$, we can solve for $\lambda$: $\lambda=\frac{c}{f}$. Given $c = 3.0\times 10^{8}\ m/s$ and $f = 6.50\times 10^{14}\ Hz$.
$\lambda=\frac{3.0\times 10^{8}\ m/s}{6.50\times 10^{14}\ Hz}$

Step2: Calculate the wavelength

$\lambda=\frac{3.0}{6.50}\times10^{8 - 14}\ m\approx4.62\times 10^{-7}\ m$

Problem 5:

Step1: First find the frequency

From $c=\lambda f$, we have $f=\frac{c}{\lambda}$. Given $c = 3.0\times 10^{8}\ m/s$ and $\lambda = 0.012\ m$.
$f=\frac{3.0\times 10^{8}\ m/s}{0.012\ m}=2.5\times 10^{10}\ Hz$

Step2: Then find the energy

Using $E = hf$, with $h = 6.63\times 10^{-34}\ J\cdot s$ and $f = 2.5\times 10^{10}\ Hz$.
$E=(6.63\times 10^{-34}\ J\cdot s)\times(2.5\times 10^{10}\ Hz)=1.6575\times 10^{-23}\ J$

Answer:

Problem 1: $4.9725\times 10^{-19}\ J$
Problem 2: $3.00\times 10^{13}\ Hz$
Problem 3: $1.0\times 10^{8}\ Hz$
Problem 4: $4.62\times 10^{-7}\ m$
Problem 5: $1.6575\times 10^{-23}\ J$