Determine the ionization energy of a hydrogen atom (in kJ/mol) if the electron is in its ground state. (Hints: Use the Rydberg equation, remember E=hc for a single H atom, and R=109678x10–2nm )

I think you meant E = hc/lambda.

Use the Rydberg equation to determine lambda. (lambda = wavelength). Use N1 = 1 and N2 = infinity
E = hc/lambda
That is Joules/atom. Multiply by 6.02 x 10^23 to convert to mols.

To determine the ionization energy of a hydrogen atom, we can use the Rydberg equation, which relates the energy of an electron transition to the wavelength of light emitted/ionization energy. The equation is as follows:

1/λ = R (1/n1^2 - 1/n2^2)

Where:
- 1/λ is the wavelength of light emitted or absorbed in inverse meters (m^-1)
- R is the Rydberg constant (1.097 x 10^7 m^-1)
- n1 and n2 are the principal quantum numbers of the initial and final energy levels, respectively.

Since we are interested in ionization energy, we want to calculate the energy transition from the ground state (n1 = 1) to the ionized state (n2 = ∞). This transition represents the complete removal of the electron from the atom.

Plugging in the values into the Rydberg equation:

1/λ = R (1/1^2 - 1/∞^2)

Simplifying the equation:

1/λ = R (1 - 0) = R

Rearranging the equation to solve for λ:

λ = 1 / R

Now we need to convert the wavelength from meters to nanometers. Since 1 m = 10^9 nm, we can convert λ as follows:

λ (nm) = (1 / R) * (10^9 nm / 1 m)

Now let's substitute the value of R given in the problem:

λ (nm) = (1 / 1.097 x 10^7 m^-1) * (10^9 nm / 1 m)

Simplifying the expression:

λ (nm) = 91.17 nm

The ionization energy (E) can be calculated using E = hc/λ, where h is Planck's constant (6.626 x 10^-34 Js) and c is the speed of light (2.998 x 10^8 m/s).

E = (6.626 x 10^-34 Js) * (2.998 x 10^8 m/s) / (91.17 x 10^-9 m)

Simplifying the expression:

E = 2.179 x 10^-18 J

To convert this energy to kJ/mol, we need to multiply it by Avogadro's number (6.022 x 10^23 mol^-1) and divide by 1000 to get the value in kJ/mol:

E (kJ/mol) = (2.179 x 10^-18 J) * (6.022 x 10^23 mol^-1) / 1000

Calculating the result:

E (kJ/mol) ≈ 1312.2 kJ/mol

Therefore, the ionization energy of a hydrogen atom in its ground state is approximately 1312.2 kJ/mol.