For emission line of atomic hydrogen from `n_(i)=8` to `n_(f)=n,` the plot of wave number `(barv)` against`((1)/(n^(2)))` will be (The Rydberg constant, `R_(H)` is in wave number unit)

Options

A. Linear with slope` R_(H)`
B. Non linear
C. Linear with slope `R_(H)`
D. Linear with intercept `R_(H)`

Explanation

To solve the problem of plotting the wave number (\( \bar{v} \)) against \( \frac{1}{n^2} \) for the transition of an electron in a hydrogen atom from an initial state \( n_i = 8 \) to a final state \( n_f = n \), we can follow these steps:

### Step 1: Write the Formula for Wave Number
The wave number (\( \bar{v} \)) for the transition between two energy levels in a hydrogen atom is given by the Rydberg formula:
\[ \bar{v} = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \]
where \( R_H \) is the Rydberg constant.

### Step 2: Substitute the Values
In our case, the initial quantum number \( n_i = 8 \) and the final quantum number \( n_f = n \). Substituting these values into the formula gives:
\[ \bar{v} = R_H \left( \frac{1}{8^2} - \frac{1}{n^2} \right) \]

### Step 3: Simplify the Expression
Calculating \( 8^2 \):
\[ 8^2 = 64 \]
Thus, we can rewrite the equation as:
\[ \bar{v} = R_H \left( \frac{1}{64} - \frac{1}{n^2} \right) \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ \bar{v} = R_H \cdot \frac{1}{64} - R_H \cdot \frac{1}{n^2} \]
This can be expressed in the form of a linear equation \( y = mx + c \):
\[ \bar{v} = -R_H \cdot \frac{1}{n^2} + R_H \cdot \frac{1}{64} \]
Here, \( y = \bar{v} \), \( x = \frac{1}{n^2} \), \( m = -R_H \), and \( c = R_H \cdot \frac{1}{64} \).

### Step 5: Identify the Characteristics of the Plot
From the equation, we can see that:
- The plot of wave number (\( \bar{v} \)) against \( \frac{1}{n^2} \) is a straight line.
- The slope of the line is negative, equal to \( -R_H \).
- The y-intercept is \( R_H \cdot \frac{1}{64} \).

### Conclusion
The plot of wave number (\( \bar{v} \)) against \( \frac{1}{n^2} \) will be a straight line with a negative slope.

### Final Answer
The answer is: **Linear with slope \(-R_H\)**.

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