To find the speed of the fluorine atom moving with the same energy as the infrared photon, we first need to calculate the energy of the infrared photon, and then use this to calculate the speed of the fluorine atom.
Step 1: Calculate the energy of an infrared photon
We use Planck's equation to do this, which is:
E = h * c / λ
where:
h is Planck's constant (6.626 * 10^-34 m^2 kg / s),
c is the speed of light (299,792,458 m/s),
and λ is the wavelength of the infrared photon (1.00 * 10^-4 meters).
Replacing the given values into the equation, we obtain:
E = (6.626 * 10^-34 ) * (299,792,458) / (1.00 * 10^-4) = 1.986424826708 * 10^-21 Joules
This is the energy of one infrared photon.
Step 2: Calculate the speed of the fluorine atom
Now we know the kinetic energy of the fluorine atom is the same as the energy of the infrared photon, we can rearrange the kinetic energy formula to find out the speed of fluorine atom.
Kinetic Energy = 1/2*m*v^2
where:
m is the mass of the fluorine atom (38.0*10^-3 kg),
v is the speed of the fluorine atom (which we are trying to find).
Rearranging the equation for v we get:
v = sqrt(2 * Kinetic Energy / m)
which gives us
v = sqrt(2 * (1.986424826708 * 10^-21) / (38.0*10^-3)) = 3.233398755023802 * 10^-10 m/s
So, the speed of a gas-phase fluorine randomly moving when having the same energy as an infrared photon is 3.23 * 10^-10 m/s.