Find out the longest wavelength of absorption line for hydrogen gas containing atoms in ground state.

Answer

To find the longest wavelength of the absorption line for hydrogen gas containing atoms in the ground state, we can follow these steps:

### Step 1: Understand the Energy Levels
In a hydrogen atom, the energy levels are quantized, and the ground state corresponds to \( n = 1 \). The first excited state is \( n = 2 \).

### Step 2: Identify the Transition
For the longest wavelength of absorption, we need to consider the transition from the first excited state (\( n = 2 \)) to the ground state (\( n = 1 \)). This transition will have the lowest energy difference, which corresponds to the longest wavelength.

### Step 3: Use the Rydberg Formula
The wavelength of the emitted or absorbed light during the transition can be calculated using the Rydberg formula:
\[ \frac{1}{\lambda} = R \cdot (1/n_1^2 - 1/n_2^2) \]
Where:
- \( R \) is the Rydberg constant (\( R = 1.097 \times 10^7 \, \text{m}^{-1} \))
- \( n_1 \) is the lower energy level (1 for ground state)
- \( n_2 \) is the higher energy level (2 for the first excited state)

### Step 4: Substitute the Values
Substituting \( n_1 = 1 \) and \( n_2 = 2 \) into the formula:
\[ \frac{1}{\lambda} = R \cdot (1/1^2 - 1/2^2) \]
\[ \frac{1}{\lambda} = R \cdot (1 - 1/4) = R \cdot (3/4) \]

### Step 5: Calculate the Wavelength
Now substituting the value of \( R \):
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \cdot \frac{3}{4} \]
\[ \frac{1}{\lambda} = 8.2275 \times 10^6 \, \text{m}^{-1} \]
To find \( \lambda \):
\[ \lambda = \frac{1}{8.2275 \times 10^6} \approx 1.215 \times 10^{-7} \, \text{m} = 121.5 \, \text{nm} \]

### Step 6: Final Answer
Thus, the longest wavelength of the absorption line for hydrogen gas in the ground state is approximately:
\[ \lambda \approx 121.5 \, \text{nm} \]