Question

The potential energy of an electron in an orbit of hydrogen atom is -6.8 eV. The de Broglie wavelength of the electron in this orbit is ($r_0$ is Bohr radius)

• A. $2\pi r_0$
• B. $4\pi r_0$
• C. $\pi r_0$
• D. $3\pi r_0$

Hint:

Remember the key relationships in the Bohr model: $E_n = -K_n = U_n/2$ and $E_n = -13.6/n^2$ eV. Also, Bohr's quantization condition can be combined with the de Broglie wavelength to give $2\pi r_n = n\lambda$, meaning the circumference of the orbit is an integer multiple of the de Broglie wavelength.

Answer 1

The Correct Option is B

In the Bohr model of the hydrogen atom, the total energy ($E_n$), potential energy ($U_n$), and kinetic energy ($K_n$) in the n-th orbit are related as follows:
$E_n = -K_n = \frac{U_n}{2}$.
We are given the potential energy $U_n = -6.8$ eV.
We can find the total energy of the electron in this orbit:
$E_n = \frac{U_n}{2} = \frac{-6.8 \text{ eV}}{2} = -3.4$ eV.
The total energy in the n-th orbit of a hydrogen atom is also given by the formula:
$E_n = -\frac{13.6}{n^2}$ eV.
Equating the two expressions for $E_n$:
$-3.4 = -\frac{13.6}{n^2} \implies n^2 = \frac{13.6}{3.4} = 4$.
So, the electron is in the second orbit, $n=2$.
According to Bohr's second postulate, the angular momentum ($L$) of the electron is quantized:
$L = mvr = n\frac{h}{2\pi}$.
The de Broglie wavelength ($\lambda$) of the electron is given by $\lambda = \frac{h}{p} = \frac{h}{mv}$.
From this, we can write $mv = \frac{h}{\lambda}$.
Substitute this into the angular momentum equation:
$\left(\frac{h}{\lambda}\right)r = n\frac{h}{2\pi}$.
$ \frac{r}{\lambda} = \frac{n}{2\pi} \implies \lambda = \frac{2\pi r}{n}$.
This is the de Broglie wavelength in terms of the radius of the n-th orbit.
For the second orbit ($n=2$), the wavelength is $\lambda = \frac{2\pi r_2}{2} = \pi r_2$.
The radius of the n-th orbit is given by $r_n = n^2 r_0$, where $r_0$ is the Bohr radius.
For $n=2$, the radius is $r_2 = 2^2 r_0 = 4r_0$.
Substituting this into the expression for the wavelength:
$\lambda = \pi (4r_0) = 4\pi r_0$.