At what temperature will the RMS speed of oxygen molecules become just sufficient for escaping from Earth’s atmosphere? Given: mass of oxygen molecule m = 2.76 × 10⁻²⁶ kg, Boltzmann constant kB = 1.38 × 10⁻²³ J K⁻¹.
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Question

At what temperature will the RMS speed of oxygen molecules become just sufficient for escaping from Earth’s atmosphere? Given: mass of oxygen molecule m = 2.76 × 10⁻²⁶ kg, Boltzmann constant kB = 1.38 × 10⁻²³ J K⁻¹.

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G1 Physics · Accepted Answer

Step 1: Escape speed from Earth is vₑ ≈ 11.2 km s⁻¹ = 1.12 × 10⁴ m s⁻¹. Step 2: Set RMS speed equal to escape speed: vRMS = √(3kBT/m) = vₑ. Step 3: Square and solve: T = mvₑ²/(3kB). Step 4: T = [(2.76 × 10⁻²⁶)(1.12 × 10⁴)²]/[3 × 1.38 × 10⁻²³] K. Step 5: T ≈ 8.36 × 10⁴ K. Final answer: (B).