CBSE · Class 12 · Physics · AtomsUsing Bohr's postulates, derive the expression for the total energy of an electron in the $n$-th orbit of a hydrogen atom. Why is this total energy negative?

Derivation of Total Energy of an Electron in $n$-th Orbit\nConsider an electron of mass $m$ and charge $e$ revolving around a nucleus of atomic number $Z$ (for hydrogen, $Z=1$) with charge $+Ze$ in a circular orbit of radius $r$ with speed $v$.

1. Kinetic Energy (K.E.): From Coulomb's law, the centripetal force is balanced by the electrostatic force: $$\frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{r^2}$$ Multiplying both sides by $\frac{r}{2}$, we get the kinetic energy ($K$): $$K = \frac{1}{2} mv^2 = \frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{2r} \quad \text{--- (Equation 1)}$$

2. Potential Energy (P.E.): The potential energy ($U$) of the electron in the electrostatic field of the nucleus is given by: $$U = \frac{1}{4\pi\varepsilon_0} \frac{(+Ze)(-e)}{r} = -\frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{r} \quad \text{--- (Equation 2)}$$

3. Total Energy (E): The total energy $E$ is the sum of kinetic energy and potential energy: $$E = K + U$$ Substituting the values from Equation (1) and Equation (2): $$E = \frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{2r} - \frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{r}$$ $$E = -\frac{1}{4\pi\varepsilon_0} \frac{Ze^2}{2r} \quad \text{--- (Equation 3)}$$

4. Substituting the Value of Radius ($r$): We know that the radius of the $n$-th Bohr orbit is: $$r = \frac{n^2 h^2 \varepsilon_0}{\pi m Z e^2}$$ Substituting this value of $r$ into Equation (3): $$E_n = -\frac{Ze^2}{4\pi\varepsilon_0 \cdot 2} \cdot \left(\frac{\pi m Z e^2}{n^2 h^2 \varepsilon_0}\right)$$ $$E_n = -\frac{m Z^2 e^4}{8 \varepsilon_0^2 n^2 h^2}$$ \nFor a hydrogen atom ($Z = 1$): $$E_n = -\frac{m e^4}{8 \varepsilon_0^2 n^2 h^2} = -\frac{13.6}{n^2} \text{ eV}$$

Why is the Total Energy Negative?\nThe total energy of the electron is negative because the electron is bound to the nucleus by attractive electrostatic forces. A negative energy signifies a bound state. To free the electron from the nucleus and take it to an infinite distance ($r = \infty$), where its potential energy and kinetic energy both become zero (total energy = zero), we must supply an external energy equal to $+13.6 \text{ eV}$ (known as ionization energy). Hence, the negative sign indicates that work must be done on the electron to remove it from the atom.