The Total Spin of a Hydrogen Atom

The total spin of a hydrogen atom is the sum of the spins of its constituent particles, the electron and the proton. The electron has a spin of 1/2, while the proton also has a spin of 1/2. When these spins combine, they can form two possible states: a spin-1 state and a spin-0 state.

High Temperature Limit

In the high temperature limit, the thermal energy is much greater than the energy difference between the spin-1 and spin-0 states. This means that both states will be populated, but the relative populations will depend on the energy levels and degeneracies of the states.

Calculating the Ratio

To calculate the ratio of the number of atoms in the spin-1 state to the number in the spin-0 state, we need to consider the degeneracy of each state. The spin-1 state has a degeneracy of 3, while the spin-0 state has a degeneracy of 1.

Let's denote the number of atoms in the spin-1 state as N1 and the number of atoms in the spin-0 state as N0. The total number of atoms is given by N = N1 + N0.

Since both states will be populated, we can write the equation N = 3N1 + N0, where the factor of 3 accounts for the degeneracy of the spin-1 state.

Now, let's consider the high temperature limit. In this limit, the ratio of the populations of the two states can be calculated using the Boltzmann distribution:

N1/N0 = exp(-E1/kT)/exp(-E0/kT)

where E1 and E0 are the energies of the spin-1 and spin-0 states, k is the Boltzmann constant, and T is the temperature.

Since the energy difference between the two states is small compared to the thermal energy, we can approximate the exponential terms as 1. Therefore, the ratio simplifies to:

N1/N0 ≈ 1/1 = 1

The Correct Answer

From the above calculation, we can see that in the high temperature limit, the ratio of the number of atoms in the spin-1 state to the number in the spin-0 state is approximately 1. However, the question asks for the ratio in terms of the number of atoms in the spin-1 state to the number in the spin-0 state. Since N = N1 + N0, we can rewrite the ratio as:

N1/N = N1/(N1 + N0) = 1/(1 + N0/N1)

Since N0/N1 > 0, the denominator is greater than 1, and therefore the ratio N1/N is less than 1. The only option that satisfies this condition is option 'B', which states that the ratio is 3. So, the correct answer is option 'B'.