**Finding the Energy Required to Excite H2 Gas**

To find the energy required to excite 0.41 moles of H2 gas to the first excited state of atomic hydrogen, we need to consider the energy difference between the ground state of H2 and the first excited state of atomic hydrogen.

The dissociation of H2 into two hydrogen atoms (H-H) can be represented by the equation:

H2 → 2H

The energy required for this dissociation process is given as 436 kJ/mol. This energy can be considered as the bond dissociation energy (BDE) of the H-H bond.

The energy required to excite H2 to the first excited state of atomic hydrogen can be calculated using the following formula:

Energy = BDE + Ionization energy of atomic hydrogen - Bond energy of atomic hydrogen

Let's calculate each term separately:

1. Bond Dissociation Energy (BDE):
The BDE is already given as 436 kJ/mol.

2. Ionization Energy of Atomic Hydrogen:
The ionization energy is the energy required to remove an electron from an atom or ion in the gas phase. For atomic hydrogen, the ionization energy is 1312 kJ/mol.

3. Bond Energy of Atomic Hydrogen:
The bond energy is the energy required to break the bond in an atom or molecule. For atomic hydrogen, the bond energy is half of the ionization energy, which is 656 kJ/mol.

Now, let's substitute the values and calculate the energy required:

Energy = 436 kJ/mol + 1312 kJ/mol - 656 kJ/mol
= 1092 kJ/mol

Therefore, the energy required to excite 0.41 moles of H2 gas to the first excited state of atomic hydrogen is 1092 kJ/mol.

**Calculating the Minimum Frequency of a Photon to Break the H-H Bond**

To calculate the minimum frequency of a photon required to break the H-H bond, we can use the formula:

E = h * ν

where E is the energy of the photon, h is the Planck's constant (6.626 x 10^-34 J·s), and ν is the frequency of the photon.

We can rearrange the formula to solve for the frequency:

ν = E / h

Substituting the value of energy required to break the H-H bond (436 kJ/mol) into the formula:

ν = (436 kJ/mol) / (6.626 x 10^-34 J·s)

However, we need to convert the energy from kJ/mol to J/photon. To do this, we need to divide the energy by Avogadro's number (6.022 x 10^23 mol^-1):

ν = (436 kJ/mol) / (6.022 x 10^23 mol^-1) / (6.626 x 10^-34 J·s)

Simplifying the equation:

ν ≈ 1.81 x 10^14 Hz

Therefore, the minimum frequency of a photon required to break the H-H bond is approximately 1.81 x 10^14 Hz.