As an electron makes a transition from an excited state to the ground state of a hydrogen like atom/

Q: Why does the kinetic energy increase when the electron drops to a lower orbit? Shouldn't it lose energy?

The electron does lose *total* energy (which is emitted as a photon). However, as it gets closer to the nucleus, the electrostatic force of attraction becomes much stronger. This stronger pull forces the electron to move faster to maintain a stable orbit, hence its kinetic energy increases.

Q: I don't understand why potential energy is negative. How can energy be less than zero?

In physics, we define the potential energy to be zero when the electron is infinitely far away from the nucleus and completely free. Because the nucleus attracts the electron, the system does work as the electron gets closer. Since it starts at zero and loses potential energy as it gets closer, the potential energy must be negative. It represents a 'bound' state.

Q: If the magnitude of the potential energy gets larger (like from 3.4 to 13.6), doesn't that mean it increases?

No, because of the negative sign! Going from $-3.4$ eV to $-13.6$ eV is a decrease in value. Think of it like a bank account: if your debt goes from $3.40 to $13.60, your overall net worth has decreased.

Q: How are KE, PE, and TE related mathematically?

For a hydrogen-like atom, they follow a strict ratio due to the Virial Theorem: $PE = -2 \times KE$, and $TE = -KE = \frac{PE}{2}$.

Q: Does this logic only apply to Hydrogen, or does it work for Helium and Lithium ions too?

It works for all hydrogen-like ions (like $\text{He}^+$, $\text{Li}^{2+}$). The formulas include a $Z^2$ term (where $Z$ is the atomic number), which scales the exact values, but the inverse-square relationship with $n$ remains exactly the same.

Question

As an electron makes a transition from an excited state to the ground state of a hydrogen like atom/ion

Options:

(A) its kinetic energy increases but potential energy and total energy decrease

(B) kinetic energy, potential energy and total energy decrease

(C) kinetic energy decreases, potential energy increases but total energy remains same

(D) kinetic energy and total energy decrease but potential energy increases

Sigma Percentile · Accepted Answer

### The Quantum Leap: A Journey to the Ground State

Imagine you are standing on the edge of a deep well. The further down you go, the deeper into the Earth's gravitational pull you fall. In the quantum world, an electron orbiting a nucleus experiences a very similar phenomenon, governed by electrostatic forces instead of gravity.

When an electron in a hydrogen-like atom makes a transition from an excited state (a higher orbit where $n > 1$) to the ground state (the lowest orbit where $n = 1$), it is essentially "falling" deeper into the electrostatic well created by the positively charged nucleus. Let's break down exactly what happens to its kinetic, potential, and total energies during this quantum leap.

### The Speed of the Fall: Kinetic Energy

Kinetic energy ($KE$) is the energy of motion. According to Bohr's model, the kinetic energy of an electron in the $n^{	ext{th}}$ orbit is given by the formula:

$$KE = \frac{13.6 Z^2}{n^2} 	ext{ eV}$$

Notice that the principal quantum number, $n$, is in the denominator. As the electron transitions to the ground state, the value of $n$ decreases. Mathematically, when the denominator of a fraction decreases, the overall value of the fraction increases.

Therefore, as $n$ decreases, the kinetic energy **increases**.

Physically, this makes perfect sense. As the electron gets closer to the positively charged nucleus, the electrostatic pull becomes stronger, causing the electron to accelerate and move faster in its tighter orbit.

### The Bound State: Potential Energy

Potential energy ($PE$) is the energy of position. For an electron bound to a nucleus, the potential energy is given by:

$$PE = -\frac{27.2 Z^2}{n^2} 	ext{ eV}$$

Here is where many students fall into a trap: **the negative sign**. The negative sign is not just a mathematical artifact; it is a profound physical statement indicating that the electron is in a bound state. It means you would have to supply energy to free the electron from the atom (bringing its potential energy to zero at an infinite distance).

As $n$ decreases, the magnitude of the fraction (the $\frac{27.2 Z^2}{n^2}$ part) increases. However, because of the negative sign, the overall value becomes *more negative*.

Think of a number line: moving from $-3.4$ to $-13.6$ is a move to the left. The value is getting smaller. Therefore, as the electron drops to a lower state, its potential energy **decreases**.

### The Grand Total: Total Energy

Total energy ($TE$) is simply the sum of kinetic and potential energy. In a bound hydrogen-like system, the total energy is always negative and is exactly half of the potential energy (or the negative of the kinetic energy). The formula is:

$$TE = -\frac{13.6 Z^2}{n^2} 	ext{ eV}$$

Following the exact same logic we used for potential energy, as $n$ decreases, the magnitude of the fraction increases, but the negative sign ensures that the overall value becomes more negative.

Thus, the total energy also **decreases**.

### Conclusion

To summarize the physics of an electron falling to the ground state:
1. It speeds up, so **Kinetic Energy increases**.
2. It falls deeper into the potential well, so **Potential Energy decreases** (becomes more negative).
3. It loses overall energy (by emitting a photon), so **Total Energy decreases** (becomes more negative).

This perfectly aligns with the statement: "its kinetic energy increases but potential energy and total energy decrease."

Animated Solution for Physics - Atoms and Nuclei: As an electron makes a transition from an excited state to the ground state of a hydrogen like atom/ion

Select Answer:

Visualized Solution

Visualizing the Transition

Kinetic Energy Formula

Analyzing Kinetic Energy

Potential Energy Formula

Analyzing Potential Energy

Total Energy Formula

Final Conclusion

The Sigma Insight: Bohr's Atomic Model and Energy Levels

The Quantum Leap

The Speed of the Fall

The Bound State

The Grand Total

Conclusion

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