Bohr Model Energy Levels in Hydrogen-Like Atoms

 

❓ Concept

Hydrogen jaisa atom dikhe…
energy difference diya ho…

👉 Samajh jao — game Z² ka hai! ⚡

Is concept ko pakad liya,
toh Bohr-model wale questions 30–40 sec mein khatam 🔥

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🖼️ Concept Image

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✍️ Short Explanation

Hydrogen-like ion problems mein calculation kam,
structure samajhna zyada important hota hai.

Bas 3 cheezein yaad rakho:

• One electron

• Bohr formula

• Z² scaling

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🔹 Step 1 — Hydrogen-Like Ion Means… (FOUNDATION 💯)**

Hydrogen-like ion ⇒
Sirf ek electron present hota hai.

Examples:

• H

• He⁺

• Li²⁺

• Be³⁺

👉 In sab par Bohr model directly apply hota hai.

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🔹 Step 2 — Bohr Energy Formula (Most Important 🔥)**

For hydrogen-like ion:

$$E_n = -\frac{13.6\,Z^2}{n^2}\quad (\text{in eV})$$

Where:

• $Z$ = atomic number

• $n$ = principal quantum number

⚡ Z² factor sabse powerful clue hai.

Energy hydrogen se compare karni ho,
toh simply multiply by $Z^2$.

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🔹 Step 3 — Excitation Level Meaning

JEE ka favourite confusion 😄

• Ground state → $n=1$
  

• 1st excited → $n=2$
  

• 2nd excited → $n=3$
  

👉 Always:

$$\text{Excitation number} + 1 = n$$

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🔹 Step 4 — Energy Difference Concept

Energy difference between two levels:

$$\Delta E = |E_{n_2} - E_{n_1}|$$

Using Bohr formula:

$$\Delta E = 13.6 Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$

📌 Important:

$$\boxed{\Delta E \propto Z^2}$$

Same transition, different atom ⇒
Energy multiplies by $Z^2$.

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🔹 Step 5 — Golden JEE Shortcut ⭐

Agar question mein energy difference:

• Hydrogen se much larger ho

• Ya exact multiple of hydrogen value ho

👉 Direct socho:

$$Z^2 = \frac{\text{Given energy}}{\text{Hydrogen energy}}$$

Then:

$$Z = \sqrt{\text{that value}}$$

🧠 Exam insight:

Options mein match Z² se hota hai,
linear Z se nahi.

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⭐ Golden Summary Box

• Hydrogen-like ⇒ one electron

• $E_n = -13.6 Z^2/n^2$
  

• Excited level: excitation + 1

• $\Delta E \propto Z^2$
  

• Compare using Z² scaling

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✅ Final Takeaway

Agar question bole:

“Hydrogen-like ion, energy difference given”

👉 Turant yaad karo:

$$\boxed{\Delta E \propto Z^2}$$

Aur hydrogen ke value se compare karo —
Z seedha mil jaata hai! 🚀