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–1 )
Use the Rydberg equation to calculate wavelength, then use E = hc/wavelength (not E = hc)
To determine the ionization energy of a hydrogen atom in its ground state, we can use the Rydberg equation and the given values for Planck's constant (h), the speed of light (c), and the Rydberg constant (R).
The Rydberg equation is given by:
1/λ = R*(1/n₁² - 1/n₂²)
Where λ is the wavelength of light emitted or absorbed, R is the Rydberg constant, and n₁ and n₂ are the principal quantum numbers. In this case, since we have a hydrogen atom in its ground state, the electron is initially in the n₁ = 1 energy level.
The ionization energy corresponds to the situation where the electron is completely removed from the atom, so it will be absorbed to infinity. This means n₂ = ∞.
Substituting the values into the equation, we have:
1/λ = R*(1/1² - 1/∞²)
However, 1/∞ is considered to be zero (approaching zero as the denominator becomes infinitely large), so we have:
1/λ = R*(1/1² - 0)
Simplifying further:
1/λ = R*(1 - 0)
1/λ = R
Now, we can rearrange the equation to solve for the wavelength:
λ = 1/R
The value of the Rydberg constant (R) is given as 109678x10–2nm–1. Therefore:
λ = 1/ (109678x10–2nm–1)
Now, we can convert the wavelength to frequency (ν) using the equation:
c = λν
Where c is the speed of light. Rearranging the equation, we have:
ν = c/λ
Substituting the values, we can calculate the frequency:
ν = (speed of light)/(1/ (109678x10–2nm–1))
Finally, using the equation:
E = hν
Where E is the energy and h is Planck's constant, we can calculate the ionization energy:
E = (Planck's constant) * (frequency)
Given that Planck's constant (h) is approximately 6.626 x 10^(-34) J·s, and the frequency (ν) from the previous calculation, we can determine the ionization energy of a hydrogen atom in its ground state (in joules). To convert it to kilojoules/mol, you would divide the value by the molar mass of hydrogen in grams (1 g/mol) and then multiply by 1000 to convert to kJ/mol.