The angular momentum J of the electron in a hydrogen atom is proportional to `n^(th)` power of r (radius of the orbit) where n is :-

Options

A. 1
B. `-1`
C. `(1)/(2)`
D. None

Answer

To solve the problem of determining the relationship between the angular momentum \( J \) of the electron in a hydrogen atom and the radius \( r \) of the orbit, we can follow these steps:

### Step 1: Write the formula for angular momentum
The angular momentum \( J \) of an electron in an orbit is given by the formula:
\[ J = mvr \]
where \( m \) is the mass of the electron, \( v \) is the velocity of the electron, and \( r \) is the radius of the orbit.

### Step 2: Relate angular momentum to quantum numbers
According to Bohr's model of the hydrogen atom, the angular momentum is quantized and can be expressed as:
\[ J = \frac{nh}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 3: Express radius in terms of quantum number
The radius of the electron's orbit in a hydrogen atom is given by:
\[ r_n = \frac{n^2}{Z} \]
where \( Z \) is the atomic number (which is 1 for hydrogen).

### Step 4: Substitute the expression for \( n \)
From the expression for the radius, we can express \( n \) in terms of \( r \):
\[ n = \sqrt{rZ} \]
Since \( Z = 1 \) for hydrogen, we have:
\[ n = \sqrt{r} \]

### Step 5: Substitute \( n \) back into the angular momentum equation
Now, substituting \( n = \sqrt{r} \) into the angular momentum equation:
\[ J = \frac{(\sqrt{r})h}{2\pi} \]

### Step 6: Simplify the equation
This simplifies to:
\[ J \propto r^{1/2} \]
Thus, the angular momentum \( J \) is proportional to \( r^{1/2} \).

### Conclusion
The value of \( n \) in the expression \( J \propto r^n \) is \( \frac{1}{2} \).

### Final Answer
The angular momentum \( J \) of the electron in a hydrogen atom is proportional to \( r^{1/2} \), therefore \( n = \frac{1}{2} \).

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