Assume a hypothetical hydrogen atom in which the potential energy between electron and proton at separation r is given by `U = [k ln r - (k/2)],` where k is a constant. For such a hypothetical hydrogen atom, calculate the radius of nth Bohr orbit and energy levels.

Explanation

To solve the problem of finding the radius of the nth Bohr orbit and the energy levels for a hypothetical hydrogen atom with the given potential energy, we will follow these steps:

### Step 1: Determine the Force from the Potential Energy
The potential energy \( U \) is given by:
\[ U = k \ln r - \frac{k}{2} \]
To find the force \( F \) between the electron and proton, we differentiate the potential energy with respect to \( r \):
\[ F = -\frac{dU}{dr} \]
Calculating the derivative:
\[ \frac{dU}{dr} = \frac{k}{r} \]
Thus, the force is:
\[ F = -\frac{k}{r} \]

### Step 2: Relate Force to Centripetal Force
According to Bohr's model, the centripetal force required to keep the electron in a circular orbit is provided by the electrostatic force of attraction:
\[ \frac{mv^2}{r} = \frac{k}{r} \]
Where \( m \) is the mass of the electron and \( v \) is its velocity. Rearranging gives:
\[ mv^2 = k \]

### Step 3: Find the Velocity of the Electron
From the equation \( mv^2 = k \), we can express the velocity \( v \):
\[ v = \sqrt{\frac{k}{m}} \]

### Step 4: Use Bohr's Second Postulate
According to Bohr's second postulate, the angular momentum \( L \) is quantized:
\[ mvr = n\frac{h}{2\pi} \]
Substituting \( v \) from Step 3:
\[ m \left(\sqrt{\frac{k}{m}}\right) r = n\frac{h}{2\pi} \]
This simplifies to:
\[ r \sqrt{mk} = n\frac{h}{2\pi} \]
Squaring both sides:
\[ r^2 mk = \left(n\frac{h}{2\pi}\right)^2 \]
Thus, the radius \( r \) of the nth Bohr orbit is:
\[ r_n = \frac{n^2 h^2}{4\pi^2 mk} \]

### Step 5: Calculate the Energy Levels
The total energy \( E \) of the electron in the nth orbit is the sum of its kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]
The kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} mv^2 = \frac{1}{2} k \]
The potential energy \( U \) at radius \( r \) is:
\[ U = k \ln r_n - \frac{k}{2} \]
Substituting \( r_n \):
\[ U = k \ln\left(\frac{n^2 h^2}{4\pi^2 mk}\right) - \frac{k}{2} \]
Thus, the total energy becomes:
\[ E = \frac{1}{2} k + k \ln\left(\frac{n^2 h^2}{4\pi^2 mk}\right) - \frac{k}{2} \]
This simplifies to:
\[ E = k \ln\left(\frac{n^2 h^2}{4\pi^2 mk}\right) \]

### Final Results
The radius of the nth Bohr orbit is:
\[ r_n = \frac{n^2 h^2}{4\pi^2 mk} \]
The energy levels are:
\[ E_n = k \ln\left(\frac{n^2 h^2}{4\pi^2 mk}\right) \]