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Sagot :
Sure, let’s address the problem step-by-step and find out the number of possible spectral lines resulting from the de-excitation of electrons in hydrogen atoms, initially in the [tex]\( n_h = 4 \)[/tex] state, as they reach the ground state [tex]\( n_1 = 1 \)[/tex].
### Step-by-Step Solution:
#### Step 1: Identify the energy states involved
Starting in the [tex]\( n_h = 4 \)[/tex] state, electrons can transition down to lower energy levels [tex]\( n = 3 \)[/tex], [tex]\( n = 2 \)[/tex], and [tex]\( n = 1 \)[/tex] (the ground state).
#### Step 2: Count all possible direct transitions
Electrons in the [tex]\( n_h = 4 \)[/tex] state can directly transition to any lower state. These direct transitions are:
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 3 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 1 \)[/tex]
#### Step 3: Consider transitions from intermediate states
Electrons that transition to intermediate states (e.g., [tex]\( n = 3 \)[/tex] or [tex]\( n = 2 \)[/tex]) can further transition to lower states. Thus, we consider the following transitions:
From [tex]\( n = 3 \)[/tex]:
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 1 \)[/tex]
From [tex]\( n = 2 \)[/tex]:
- From [tex]\( n = 2 \)[/tex] to [tex]\( n = 1 \)[/tex]
### Step 4: Diagram all possible pathways and find unique transitions
To ensure all pathways are included, let’s list all the transitions:
1. [tex]\( 4 \rightarrow 3 \)[/tex]
2. [tex]\( 4 \rightarrow 2 \)[/tex]
3. [tex]\( 4 \rightarrow 1 \)[/tex]
4. [tex]\( 3 \rightarrow 2 \)[/tex]
5. [tex]\( 3 \rightarrow 1 \)[/tex]
6. [tex]\( 2 \rightarrow 1 \)[/tex]
Summarizing, we have the following unique transitions:
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 3 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 1 \)[/tex]
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 1 \)[/tex]
- From [tex]\( n = 2 \)[/tex] to [tex]\( n = 1 \)[/tex]
### Step 5: Number of spectral lines
The number of different spectral lines corresponds to the number of unique transitions:
- The total number of transitions is 6.
Therefore, there are 6 possible spectral lines that can appear in the emission spectrum as a result of electrons in the [tex]\( n_h = 4 \)[/tex] state reaching the ground state [tex]\( n_1 = 1 \)[/tex].
### Conclusion
When electrons de-excite from [tex]\( n_h = 4 \)[/tex] to the ground state [tex]\( n_1 = 1 \)[/tex] in hydrogen atoms, there can be a total of 6 possible spectral lines appearing in the emission spectrum.
The transitions can be diagrammed as follows:
- Direct transitions: [tex]\( 4 \rightarrow 3 \)[/tex], [tex]\( 4 \rightarrow 2 \)[/tex], [tex]\( 4 \rightarrow 1 \)[/tex]
- From intermediate states: [tex]\( 3 \rightarrow 2 \)[/tex], [tex]\( 3 \rightarrow 1 \)[/tex], [tex]\( 2 \rightarrow 1 \)[/tex]
These are represented by the transitions:
[tex]\[ (4, 3), (4, 2), (4, 1), (3, 2), (3, 1), (2, 1) \][/tex]
Ultimately, the solution gives us the following result:
- Transitions: [tex]\([(4, 3), (4, 2), (4, 1), (3, 2), (3, 1), (2, 1)]\)[/tex]
- Number of spectral lines: [tex]\(6\)[/tex].
### Step-by-Step Solution:
#### Step 1: Identify the energy states involved
Starting in the [tex]\( n_h = 4 \)[/tex] state, electrons can transition down to lower energy levels [tex]\( n = 3 \)[/tex], [tex]\( n = 2 \)[/tex], and [tex]\( n = 1 \)[/tex] (the ground state).
#### Step 2: Count all possible direct transitions
Electrons in the [tex]\( n_h = 4 \)[/tex] state can directly transition to any lower state. These direct transitions are:
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 3 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 1 \)[/tex]
#### Step 3: Consider transitions from intermediate states
Electrons that transition to intermediate states (e.g., [tex]\( n = 3 \)[/tex] or [tex]\( n = 2 \)[/tex]) can further transition to lower states. Thus, we consider the following transitions:
From [tex]\( n = 3 \)[/tex]:
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 1 \)[/tex]
From [tex]\( n = 2 \)[/tex]:
- From [tex]\( n = 2 \)[/tex] to [tex]\( n = 1 \)[/tex]
### Step 4: Diagram all possible pathways and find unique transitions
To ensure all pathways are included, let’s list all the transitions:
1. [tex]\( 4 \rightarrow 3 \)[/tex]
2. [tex]\( 4 \rightarrow 2 \)[/tex]
3. [tex]\( 4 \rightarrow 1 \)[/tex]
4. [tex]\( 3 \rightarrow 2 \)[/tex]
5. [tex]\( 3 \rightarrow 1 \)[/tex]
6. [tex]\( 2 \rightarrow 1 \)[/tex]
Summarizing, we have the following unique transitions:
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 3 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 4 \)[/tex] to [tex]\( n = 1 \)[/tex]
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 2 \)[/tex]
- From [tex]\( n = 3 \)[/tex] to [tex]\( n = 1 \)[/tex]
- From [tex]\( n = 2 \)[/tex] to [tex]\( n = 1 \)[/tex]
### Step 5: Number of spectral lines
The number of different spectral lines corresponds to the number of unique transitions:
- The total number of transitions is 6.
Therefore, there are 6 possible spectral lines that can appear in the emission spectrum as a result of electrons in the [tex]\( n_h = 4 \)[/tex] state reaching the ground state [tex]\( n_1 = 1 \)[/tex].
### Conclusion
When electrons de-excite from [tex]\( n_h = 4 \)[/tex] to the ground state [tex]\( n_1 = 1 \)[/tex] in hydrogen atoms, there can be a total of 6 possible spectral lines appearing in the emission spectrum.
The transitions can be diagrammed as follows:
- Direct transitions: [tex]\( 4 \rightarrow 3 \)[/tex], [tex]\( 4 \rightarrow 2 \)[/tex], [tex]\( 4 \rightarrow 1 \)[/tex]
- From intermediate states: [tex]\( 3 \rightarrow 2 \)[/tex], [tex]\( 3 \rightarrow 1 \)[/tex], [tex]\( 2 \rightarrow 1 \)[/tex]
These are represented by the transitions:
[tex]\[ (4, 3), (4, 2), (4, 1), (3, 2), (3, 1), (2, 1) \][/tex]
Ultimately, the solution gives us the following result:
- Transitions: [tex]\([(4, 3), (4, 2), (4, 1), (3, 2), (3, 1), (2, 1)]\)[/tex]
- Number of spectral lines: [tex]\(6\)[/tex].
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