Calculate the energy difference (E) for the transition from n = 1 to n = 6 energy levels of hydrogen per 1 mol of H atoms. (Report your answer to at least 3 significant figures.)
To calculate the energy difference for the transition from n = 1 to n = 6 energy levels of hydrogen, we can use the formula for the energy of a hydrogen atom:
E = -13.6 * (Z^2 / n^2) eV
where:
- E is the energy in electron volts (eV)
- Z is the atomic number of the nucleus (which is 1 for hydrogen)
- n is the energy level
Using this formula, we can calculate the energy difference between the n = 1 and n = 6 energy levels.
For n = 1:
E1 = -13.6 * (1^2 / 1^2) = -13.6 eV
For n = 6:
E6 = -13.6 * (1^2 / 6^2) = -13.6 * (1/36) = -0.3778 eV
To find the energy difference, we subtract the lower energy level from the higher energy level:
E = E6 - E1 = -0.3778 eV - (-13.6 eV) = 13.2222 eV
Now, to calculate the energy difference per 1 mol of H atoms, we can use Avogadro's number (6.022 * 10^23) to convert from per atom to per mole:
E_mol = E * N_A
where:
- E_mol is the energy difference per 1 mol of H atoms
- N_A is Avogadro's number (approximately 6.022 * 10^23)
Calculating E_mol:
E_mol = 13.2222 eV * (6.022 * 10^23) = 7.9548 * 10^24 eV/mol
Finally, we can convert the energy from electron volts to joules.
1 eV = 1.60218 * 10^-19 J
E_mol_J = E_mol * (1.60218 * 10^-19 J / 1 eV)
Calculating E_mol_J:
E_mol_J = 7.9548 * 10^24 eV * (1.60218 * 10^-19 J / 1 eV) = 1.2734 * 10^6 J/mol
Therefore, the energy difference for the transition from n = 1 to n = 6 energy levels of hydrogen per 1 mol of H atoms is approximately 1.2734 * 10^6 J/mol.