# Question 810cb

Jul 12, 2017

U= KE + PE gives negative.

#### Explanation:

$P E = - G M \frac{m}{R}$

$K E = \frac{1}{2} G M \frac{m}{R}$

Clearly, PE = -2KE

So, $U = P E + K E$
$U = - G M \frac{m}{2 R}$

So, total energy is always negative.

Hope, you will understand. Thank you!

Jul 12, 2017

The energy is negative essentially because energy must be supplied to the electron to remove it from the atom.

#### Explanation:

This is all written ignoring the fact that electrons do not in fact orbit atomic nuclei. If you want brief introduction to why the orbital model is not accurate see this answer.

Mathematically speaking
In the orbital model of the atom electrostatic forces provide the centripetal force to keep electrons orbiting the much more massive atomic nucleus (mass of electron = $\frac{1}{1836}$ mass of proton).

Centripetal force: ${F}_{c} = \frac{m {v}^{2}}{r}$

Electrostatic force: ${F}_{E} = \frac{k {q}_{1} {q}_{2}}{r} ^ 2$

Combine ① and ②: $\frac{m {v}^{2}}{r} = \frac{k {q}_{1} {q}_{2}}{r} ^ 2$
⇒ mv^2 = (kq_1q_2)/r

In the case of the hydrogen atom the proton in the nucleus is charged and the orbital electron is charged. Both have a charge of magnitude of e (the elementary charge). Sign is not important here since we are only interested in the magnitude of the force.

Substitute for ${q}_{1}$ and ${q}_{2}$ in the above equation:
⇒ mv^2 = (ke^2)/r

Total energy is the sum of kinetic and potential energies:
$U = K E + P E$

Kinetic energy: KE = 1/2mv²
Now substitute using equation ③: $K E = \frac{1}{2} \frac{k {e}^{2}}{r}$
That's our equation for KE.

Potential energy is calculated from the electric potential energy of the electron. In this case it represents how much energy is required to completely remove the electron from the atom.
$P E = \frac{k Q q}{r}$
Again both charges are the elementary charge, e .
⇒ PE = (ke^2) / r

Here we must introduce the sign. The electron (positive) is attracted to the nucleus (negative). So to remove it from the atom (as per the above statement) we must supply energy, so PE is negative. (KE was positive because it is simply energy that the electron has due to its motion.
⇒ PE = -(ke^2) / r

Now considering total energy:
$U = P E + K E = - \frac{k {e}^{2}}{r} + \frac{1}{2} \frac{k {e}^{2}}{r}$
⇒ U = -1/2 (ke^2)/r#