One cubic metre of atomic hydrogen at 0​​​^oC and atmospheric pressure contains about 2.70×10²⁵ atoms.Energy value of a hydrogen atom in a stationary state is -13.6/n² (eV).Use Boltzmann’s factor to find the number of atoms in the first excited state at 0^oC.

To determine the number of hydrogen atoms in the first excited state at 0 °C using Boltzmann's factor, we need to follow a few logical steps. Let's break this down into manageable parts.

Understanding the Energy Levels of Hydrogen

The energy of a hydrogen atom in a stationary state is given by the formula:

E_n = -13.6/n² (eV)

Here, n represents the principal quantum number. For hydrogen:

• The ground state (n=1) has an energy of E₁ = -13.6 eV.
• The first excited state (n=2) has an energy of E₂ = -3.4 eV.

Applying Boltzmann's Factor

Boltzmann's factor describes the relative probability of a system being in a particular energy state at a given temperature. It is expressed as:

P_n = (g_n * e^(-E_n / kT))

Where:

• P_n is the probability of finding the system in state n.
• g_n is the degeneracy of the state (for hydrogen, g₁ = 2 for n=1 and g₂ = 8 for n=2).
• E_n is the energy of the state.
• k is Boltzmann's constant (approximately 8.617 x 10^-5 eV/K).
• T is the temperature in Kelvin.

Calculating the Probabilities

At 0 °C, the temperature in Kelvin is:

T = 273.15 K

Now, we can calculate the probabilities for the ground state and the first excited state:

Ground State (n=1)

For the ground state:

E₁ = -13.6 eV

The probability is:

P₁ = g₁ * e^(-E₁ / kT) = 2 * e^(13.6 / (8.617 x 10^-5 * 273.15))

First Excited State (n=2)

For the first excited state:

E₂ = -3.4 eV

The probability is:

P₂ = g₂ * e^(-E₂ / kT) = 8 * e^(3.4 / (8.617 x 10^-5 * 273.15))

Finding the Number of Atoms in the Excited State

Now, we need to find the ratio of the number of atoms in the excited state to the total number of atoms:

N₂ = N * (P₂ / (P₁ + P₂))

Where N is the total number of hydrogen atoms, which is given as:

N = 2.70 x 10²⁵ atoms

Putting It All Together

After calculating the probabilities, you can substitute them into the equation for N₂ to find the number of atoms in the first excited state. The calculations will yield a very small number of atoms in the excited state compared to the ground state, reflecting the fact that at low temperatures, most atoms occupy the lowest energy state.

This approach illustrates how quantum mechanics and statistical mechanics intersect, providing insight into the behavior of particles at the atomic level. If you have any further questions or need clarification on any of these steps, feel free to ask!