Consider an electron in the nth orbit of a hydrogen atom in the Bohr model. The circumference of the orbit can be expressed in terms of the de Broglie wavelength `lambda` of that electron as

Options

A. `(0.529)nlambda`
B. `sqrt(nlambda)`
C. `(13.6)lambda`
D. `nlambda`

Answer

To solve the problem of expressing the circumference of the nth orbit of a hydrogen atom in terms of the de Broglie wavelength (λ) of the electron, we can follow these steps:

### Step 1: Understand the de Broglie wavelength
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle. For an electron, the momentum \( p \) can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 2: Write the expression for the circumference of the orbit
The circumference \( C \) of the nth orbit of a hydrogen atom is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the nth orbit.

### Step 3: Use the Bohr model to relate angular momentum and radius
According to the Bohr model, the angular momentum \( L \) of the electron in the nth orbit is quantized and given by:
\[ L = mvr = \frac{nh}{2\pi} \]
where \( n \) is the principal quantum number.

### Step 4: Express the radius in terms of de Broglie wavelength
From the angular momentum equation, we can express \( r \) as:
\[ r = \frac{nh}{2\pi mv} \]
Now, we can substitute the expression for momentum \( p = mv \) into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} \]
Thus, we can write:
\[ mv = \frac{h}{\lambda} \]
Substituting this back into the expression for \( r \):
\[ r = \frac{nh}{2\pi \left(\frac{h}{\lambda}\right)} = \frac{n\lambda}{2\pi} \]

### Step 5: Substitute \( r \) back into the circumference formula
Now, substituting \( r \) into the circumference formula:
\[ C = 2\pi r = 2\pi \left(\frac{n\lambda}{2\pi}\right) \]
The \( 2\pi \) terms cancel out:
\[ C = n\lambda \]

### Final Answer
Thus, the circumference of the nth orbit of a hydrogen atom can be expressed as:
\[ C = n\lambda \]