The first ionization energy of a carbon atom is 1.81 aJ. What is the frequency and wavelength, in nanometers, of photons capable of just ionizing carbon atoms?

v=______s^(-1)

lambda =_____nm

Assuming an ionization efficiency of 47.0%, how many such photons are needed to ionize 1.00 × 1016 atoms?

=_______protons

What's 1.81 aJ? Do you mean kJ so IP = 1810 J.

Then 1810 = hc/wavelength. Solve for wavelength in meters and convert to nm.
c = freq x wavelength will let you solve for freq.

1810 J/atom x 1E16 atoms x 0.47 = ?

To find the frequency and wavelength of photons capable of just ionizing carbon atoms, we can start by using the equation:

E = h * v

where E is the energy required for ionization, h is the Planck's constant (6.626 x 10^(-34) J * s), and v is the frequency of the photons.

Given that the first ionization energy of a carbon atom is 1.81 aJ, we can convert it to joules:

1.81 aJ = 1.81 x 10^(-18) J

Now we can rearrange the equation to solve for the frequency:

v = E / h
v = (1.81 x 10^(-18) J) / (6.626 x 10^(-34) J * s)

Calculating this, we find:

v ≈ 2.74 x 10^15 s^(-1)

Next, we can use the equation:

c = λ * v

where c is the speed of light (3.00 x 10^8 m/s) and λ is the wavelength of the photons.

We can rearrange this equation to solve for the wavelength:

λ = c / v
λ = (3.00 x 10^8 m/s) / (2.74 x 10^15 s^(-1))

Converting the wavelength from meters to nanometers (1 nm = 1 x 10^(-9) m), we find:

λ ≈ 109 nm

So the frequency of the photons capable of just ionizing carbon atoms is approximately 2.74 x 10^15 s^(-1), and their wavelength is approximately 109 nm.

Now, let's calculate the number of photons needed to ionize 1.00 × 10^16 carbon atoms, assuming an ionization efficiency of 47.0%.

Since the ionization efficiency is 47.0%, we can multiply the total number of carbon atoms by this percentage to find the number of atoms that will actually be ionized:

Number of ionized atoms = (1.00 × 10^16 atoms) * (0.470)

Next, we can calculate the total number of photons needed to ionize these atoms. Since each ionized atom will require one photon, the number of photons needed will be the same as the number of ionized atoms:

Number of photons needed = Number of ionized atoms

Therefore, the number of photons needed to ionize 1.00 × 10^16 carbon atoms, assuming an ionization efficiency of 47.0%, is approximately 4.70 × 10^15 photons.

Finally, you mentioned "protons" in your question, but it is unclear how protons relate to the ionization process. If you meant to ask about the number of protons in the ionized carbon atoms, the answer would depend on the specific ionization process and the resulting ion.

To find the frequency and wavelength of photons capable of just ionizing carbon atoms, we can use the equation:

E = hv

Where E is the energy of the photon, h is Planck's constant, and v is the frequency of the photon.

First, let's convert the ionization energy from aJ (attojoules) to joules (J) to match the unit of energy in the equation.

1 aJ = 1 x 10^(-18) J

So the ionization energy of carbon is:

E = 1.81 x 10^(-18) J

Now, let's calculate the frequency of the photons:

E = hv

v = E / h

h is Planck's constant and its value is 6.626 x 10^(-34) J∙s.

v = (1.81 x 10^(-18) J) / (6.626 x 10^(-34) J∙s)

v ≈ 2.74 x 10^15 s^(-1)

Now, to find the wavelength of the photons, we can use the equation:

c = λv

Where c is the speed of light, λ is the wavelength of the photon, and v is the frequency of the photon.

The speed of light, c, is approximately 3.0 x 10^8 m/s.

Let's convert this to nanometers (nm):

1 m = 1 x 10^9 nm

So, the speed of light is:

c = 3.0 x 10^8 m/s = 3.0 x 10^17 nm/s

Now, let's calculate the wavelength:

c = λv

λ = c / v

λ = (3.0 x 10^17 nm/s) / (2.74 x 10^15 s^(-1))

λ ≈ 109.49 nm

Therefore, the frequency is approximately 2.74 x 10^15 s^(-1) and the wavelength is approximately 109.49 nm.

Now, let's move on to the second part of the question.

Assuming an ionization efficiency of 47.0%, we can calculate the number of photons needed to ionize 1.00 × 10^16 carbon atoms.

First, let's find the number of atoms that need to be ionized:

Number of carbon atoms = 1.00 × 10^16 atoms

Now, let's calculate the number of photons needed:

Number of photons = Number of carbon atoms / Ionization efficiency

Number of photons = (1.00 × 10^16 atoms) / 0.470

Number of photons ≈ 2.13 × 10^16 photons

Finally, to find the number of protons in this case, we need to multiply the number of photons by the charge of each proton. Each ionized carbon atom loses one electron, which corresponds to one positive charge, which is the charge of a proton.

Number of protons = Number of photons

Number of protons ≈ 2.13 × 10^16 protons

So, the number of protons needed to ionize 1.00 × 10^16 carbon atoms is approximately 2.13 × 10^16 protons.