A: If a beam of photons of energy 10.0 eV each is incident on a sample of hydrogen gas containing all atoms in the ground state, then the beam of the photons is completely transmitted through the gas without absorption.
R: The minimum energy required by an electron to make a transition to an excited state is `10.2 eV`.`

Options

A. If both Assertion & Reason are true and the reason is the correct explantion of the assertion , then mark (1)
B. if both Assertion & Reason are true but the reason is not the correct explantion of the assertion , then mark (2)
C. If Assertion is true statement but Reason is false, then mark (3)
D. If both Assertion and Reason are false statements, then mark (4)

Answer

To analyze the assertion and reason provided in the question, we will break down the concepts involved in the interaction of photons with hydrogen atoms in their ground state.

### Step-by-Step Solution:

1. **Understanding the Ground State of Hydrogen:**
- The ground state energy of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the ground state (n=1), the energy is:
\[ E_1 = -13.6 \, \text{eV} \]

2. **Energy Levels of Hydrogen:**
- The first excited state (n=2) energy is:
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]
- The energy difference between the ground state and the first excited state is:
\[ \Delta E = E_2 - E_1 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \]

3. **Photon Energy Incident on Hydrogen:**
- The energy of the incident photons is given as 10.0 eV.
- To excite an electron from the ground state to the first excited state, it requires a minimum energy of 10.2 eV.

4. **Comparison of Photon Energy and Required Energy:**
- Since the energy of the photons (10.0 eV) is less than the minimum energy required (10.2 eV) for an electron to transition to an excited state, the photons will not be absorbed by the hydrogen atoms.

5. **Conclusion on Assertion:**
- The assertion states that the beam of photons is completely transmitted through the gas without absorption, which is correct because the photon energy is insufficient to excite the electrons.

6. **Conclusion on Reason:**
- The reason states that the minimum energy required by an electron to make a transition to an excited state is 10.2 eV, which we have calculated and confirmed to be true.

### Final Answer:
Both the assertion (A) and the reason (R) are correct. The assertion is true because the photons do not have enough energy to be absorbed, and the reason is true as it correctly states the minimum energy required for excitation.