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

Explanation

To solve the problem, we need to analyze the emission lines of atomic hydrogen as it transitions from an initial energy level \( n_i = 8 \) to a final energy level \( n_f = n \). We will derive the relationship between the wave number \( \bar{v} \) and \( \frac{1}{n^2} \).

### Step 1: Understand the formula for wave number
The wave number \( \bar{v} \) is related to the Rydberg formula for hydrogen:
\[ \bar{v} = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \]
where \( R_H \) is the Rydberg constant.

### Step 2: Substitute the initial and final states
Given that \( n_i = 8 \) and \( n_f = n \), we can substitute these values into the formula:
\[ \bar{v} = R_H \left( \frac{1}{8^2} - \frac{1}{n^2} \right) \]
Calculating \( 8^2 \):
\[ 8^2 = 64 \]
Thus, we have:
\[ \bar{v} = R_H \left( \frac{1}{64} - \frac{1}{n^2} \right) \]

### Step 3: Rearranging the equation
We can express this equation in terms of \( \frac{1}{n^2} \):
\[ \bar{v} = R_H \left( \frac{1}{64} \right) - R_H \left( \frac{1}{n^2} \right) \]
This can be rewritten as:
\[ \bar{v} = -R_H \left( \frac{1}{n^2} \right) + R_H \left( \frac{1}{64} \right) \]

### Step 4: Identify the linear relationship
This equation is in the form of \( y = mx + c \), where:
- \( y \) is \( \bar{v} \)
- \( m \) (the slope) is \( -R_H \)
- \( x \) is \( \frac{1}{n^2} \)
- \( c \) (the y-intercept) is \( R_H \left( \frac{1}{64} \right) \)

### Step 5: Conclusion about the plot
Since the relationship between \( \bar{v} \) and \( \frac{1}{n^2} \) is linear, we conclude that the plot of wave number \( \bar{v} \) against \( \frac{1}{n^2} \) will be a straight line with a negative slope.

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

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