In a hydrogen atom, if the radius of the orbit of the electron is doubled, then its energy will ...
>> Increase by a factor of 2??
No, increasing the radius of the electron's orbit in a hydrogen atom will actually decrease its energy. The energy of the electron in a hydrogen atom is inversely proportional to the square of the radius of its orbit. Therefore, doubling the radius of the orbit will result in the electron's energy decreasing by a factor of 4, not increasing.
To understand how the energy of an electron in a hydrogen atom relates to the radius of its orbit, we need to consider the concept of Bohr's model of the hydrogen atom.
According to Bohr's model, the energy of an electron in a hydrogen atom depends on the principle quantum number (n) and is inversely proportional to the square of the radius of its orbit. Mathematically, the energy of the electron (En) is given by the equation:
En = -13.6 eV / n^2
Where:
- En is the energy of the electron
- -13.6 eV is a constant relating to the energy of the hydrogen atom
- n is the principle quantum number
Now, let's look at the scenario you presented. If the radius of the electron's orbit is doubled, it means that the new radius (r') would be twice the original radius (r). Mathematically, r' = 2r.
From the equation mentioned earlier, we can see that the energy (En) is inversely proportional to the square of the radius (r). As a result, if the radius is doubled, the denominator (r^2) becomes 4 times larger (2^2 = 4).
So, with the radius doubled, the energy of the electron would be decreased by a factor of 4. This means that the energy will decrease by a factor of 4, not increase by a factor of 2 as you initially suggested.
Therefore, the correct statement would be: If the radius of the orbit of the electron in a hydrogen atom is doubled, then its energy will decrease by a factor of 4.
Actually "Actually, it is not possible for the radius of the orbit to double"
is one of the options, so is that right for the reason you stated?
The radius of a Bohr orbit increases with the square of the quantum number, n. So the principal quantum number would have to increase by a factor of sqrt2. This will not be possible since the quantum number must remain an integer.
Your teacher may not be aware of this.
In general, the total energy of a Bohr orbit becomes larger (less nagative) when the radius of the orbit increases. A totally detached electron (infinite radius) has an energy of zero.