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 )
_____ kJ/mol
Thanks :)
1/wavelength = R(1/n1^2 - 1/n2^2)
n1 = 1
n2 = infinity which makes 1/n2^2 = 0 so you have
1/wavelength = R(1)
Calculate wavelength.
Then E = hc/wavelength.
Solve for E in J. That will be for 1 atom. Multiply by 6.022E23 to convert to J/mol and convert that to kJ/mol.
1.312E03
To determine the ionization energy of a hydrogen atom in its ground state, we can use the Rydberg equation. The formula for the ionization energy of hydrogen is given by:
E = -R_H*(1/n²)
where E is the ionization energy, R_H is the Rydberg constant for hydrogen, and n is the principal quantum number of the energy level the electron is transitioning from.
In this case, since the electron is in its ground state, it is transitioning from the first energy level, which has a principal quantum number of n=1.
The Rydberg constant for hydrogen, R_H, is equal to 109678 x 10^2 nm^-1.
Now we can substitute these values into the formula:
E = -(109678 x 10^2 nm^-1)*(1/1²)
Simplifying the equation:
E = -(109678 x 10^2 nm^-1)
Converting nm to m:
E = -(109678 x 10^2 m^-1)
Finally, we can convert the energy to kJ/mol by using the relationship E = (2.1799 x 10^-18 J)*(6.022 x 10^23 mol^-1)/(1000 J kJ^-1):
E = -(109678 x 10^2 m^-1) * (2.1799 x 10^-18 J)*(6.022 x 10^23 mol^-1)/(1000 J kJ^-1)
Calculating this expression:
E = -1312 kJ/mol
Therefore, the ionization energy of a hydrogen atom in its ground state is approximately 1312 kJ/mol.
To determine the ionization energy of a hydrogen atom in its ground state, we can use the Rydberg equation. The Rydberg equation is given by:
1/λ = R * (1/n1^2 - 1/n2^2)
Where:
- λ is the wavelength of light emitted or absorbed
- R is the Rydberg constant (given as 109678 × 10^2 nm^-1)
- n1 and n2 are integers representing the energy levels
In the case of ionization energy, we are concerned with the electron moving from the first energy level (n1) to infinity (n2).
Setting n1 to 1 and n2 to infinity, the equation becomes:
1/λ = R * (1/1^2 - 1/infinity^2)
Since 1/infinity^2 equals zero, the equation simplifies to:
1/λ = R * (1 - 0)
Simplifying further:
1/λ = R
Now, we can use the relation E = hc/λ, where E is the energy of a photon, h is Planck's constant (6.626 × 10^-34 J·s), and c is the speed of light (3 × 10^8 m/s).
E = hc/λ
E = (6.626 × 10^-34 J·s) (3 × 10^8 m/s) / λ
Since we are interested in the ionization energy in kJ/mol, we need to convert the unit of energy from joules to kilojoules. There are 1000 joules in 1 kilojoule.
E = (6.626 × 10^-34 J·s) (3 × 10^8 m/s) / λ / 1000
E = (19.878 × 10^-26 J·m) / λ / 1000
The wavelength of light emitted or absorbed can be calculated using the relation:
λ = c / frequency
The frequency of light emitted or absorbed for ionization is the threshold frequency, ν0. We can substitute this into the equation:
E = (19.878 × 10^-26 J·m) / (c / ν0) / 1000
E = (19.878 × 10^-26 J·m) ν0 / c / 1000
Now we can plug in the values:
E = (19.878 × 10^-26 J·m) (3 × 10^8 s^-1) / (2.998 × 10^8 m/s) / 1000
After calculating this expression, we will get the ionization energy of the hydrogen atom in joules. To convert it to kJ/mol, we need to multiply the result by Avogadro's constant (6.022 × 10^23 mol^-1).