How much faster are the hot molecules?

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Problem: How much faster are the hot molecules?
A gas warms from 200 K to 800 K without changing its molecular species.
Question: Find the ratio of final to initial rms speed.
Givens:
- T₁ = 200 K
- T₂ = 800 K
Assumptions: Ideal gas at equilibrium; same molecular species; temperatures are absolute kelvin values.
Choices (LaTeX):
A. 4
B. 1/2
C. 16
D. 2

Ask for my approach first. Help me distinguish a physical misconception from an algebra error without revealing which choice is correct.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.