The wave number of the first emission line in the Balmer series of H - Spectrum is `(n)/(36)R`. The value of 'n' is. (R = Rydberg constant):

Answer

To find the value of 'n' for the first emission line in the Balmer series of the hydrogen spectrum, we can follow these steps:

### Step 1: Understand the Balmer Series
The Balmer series corresponds to transitions of electrons in a hydrogen atom from higher energy levels (n2) to the second energy level (n1 = 2). The first emission line in the Balmer series occurs when the electron transitions from n2 = 3 to n1 = 2.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wave number (ṽ) of the spectral lines is given by:
\[ \tilde{\nu} = R \cdot Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Where:
- \( R \) is the Rydberg constant,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n_1 \) is the lower energy level,
- \( n_2 \) is the higher energy level.

### Step 3: Assign Values
For the first emission line in the Balmer series:
- \( n_1 = 2 \)
- \( n_2 = 3 \)

### Step 4: Substitute Values into the Formula
Substituting the values into the Rydberg formula:
\[ \tilde{\nu} = R \cdot 1^2 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \]
\[ \tilde{\nu} = R \left( \frac{1}{4} - \frac{1}{9} \right) \]

### Step 5: Calculate the Difference
To calculate \( \frac{1}{4} - \frac{1}{9} \), we need a common denominator:
- The least common multiple of 4 and 9 is 36.
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36} \]
Thus,
\[ \frac{1}{4} - \frac{1}{9} = \frac{9}{36} - \frac{4}{36} = \frac{5}{36} \]

### Step 6: Substitute Back into the Wave Number Equation
Now substituting back into the equation:
\[ \tilde{\nu} = R \cdot \frac{5}{36} \]

### Step 7: Relate to Given Wave Number
We are given that the wave number of the first emission line in the Balmer series is \( \frac{n}{36}R \). Thus, we can equate:
\[ R \cdot \frac{5}{36} = \frac{n}{36}R \]

### Step 8: Solve for 'n'
Since \( R \) is common on both sides, we can cancel it out:
\[ \frac{5}{36} = \frac{n}{36} \]
This implies:
\[ n = 5 \]

### Final Answer
The value of 'n' is **5**.

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