The ratio of de-Broglie wavelength of electron and proton moving with the same speed is about — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - de-Broglie Wavelength Formula According to de-Broglie's wave-particle duality: $$\lambda = \frac{h}{p} = \frac{h}{m \cdot v}$$ ### Step 2 - Proportionality at Equal Speed Since both particles move at the same speed $v$: $$\lambda \propto \frac{1}{m}$$ The ratio of wavelengths: $$\frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e}$$ ### Step 3 - Substitute Known Mass Ratio $$m_p \approx 1836 \cdot m_e \implies \frac{m_p}{m_e} \approx 1836$$ $$\frac{\lambda_e}{\lambda_p} = 1836 \implies \lambda_e : \lambda_p \approx \boxed{1836 : 1}$$ ### Step 4 - Evaluation of Options * **Option (A) 1836:1:** Correct — electron has much larger wavelength due to its much smaller mass. * **Option (B) 1:1836:** Incorrect — this is the ratio of proton to electron wavelength ($\lambda_p:\lambda_e$), which is the inverse. * **Option (C) 1:1:** Incorrect — would require equal masses. * **Option (D) 1:2:** Incorrect — no physical basis. $$\text{Correct Option: } \boxed{\text{A}}$$