The first ionization energy of a hydrogen atom is 2.18 aJ. Assuming the ionization efficiency is 82%, how many such photons are needed to 1.00 * 10^16 atoms?
To find out how many photons are needed to ionize a certain number of hydrogen atoms, we need to calculate the number of atoms that will be ionized and then divide that by the ionization efficiency.
First, let's calculate the number of atoms that will be ionized. Given that we have 1.00 x 10^16 atoms, we'll use this value to find the number of ionizations.
Number of ionizations = (Number of atoms) x (Ionization efficiency)
Number of ionizations = (1.00 x 10^16 atoms) x (0.82)
Now, let's calculate the total energy required for the ionizations. We'll multiply the number of ionizations by the ionization energy of a hydrogen atom.
Total energy required = (Number of ionizations) x (Ionization energy of hydrogen atom)
Here's how to calculate it:
Total energy required = (1.00 x 10^16 atoms x 0.82) x (2.18 aJ)
To simplify the calculation, we can convert the ionization energy to joules:
2.18 aJ = 2.18 x 10^-18 J
Now we can calculate the total energy required:
Total energy required = (1.00 x 10^16 atoms x 0.82) x (2.18 x 10^-18 J)
Finally, we need to calculate the number of photons required to provide this energy. Since each photon carries energy, we can find the number of photons by dividing the total energy required by the energy carried by one photon. The energy of a single photon can be calculated using the equation:
Energy of a single photon = Planck's constant x frequency
Now, the frequency of a photon can be found using the equation:
Frequency = Speed of light / Wavelength
For hydrogen, the corresponding wavelength for ionization can be found using the Rydberg formula:
1/λ = R_H * (1/n_f^2 - 1/n_i^2)
where R_H is the Rydberg constant for hydrogen (1.097 x 10^7 m^-1), n_f is the final energy level, and n_i is the initial energy level. Since we're dealing with ionization from the ground state (n_i = 1), the final energy level will be n_f = ∞.
Plugging in the values, we can calculate the wavelength and then use it to find the frequency. Finally, we can find the energy of a single photon.
Once we have the energy of a single photon, we can divide the total energy required by the energy of one photon to find the number of photons.
I hope this explanation helps you understand how to calculate the number of photons needed to ionize a certain number of hydrogen atoms!
Is that 2.18 J/atom?
2.18/0.82 = 2.66 J/atom required.
2.66 x 1E16 = ?