If the root mean square (rms) speed of the molecules in a gas is to be tripled, to what temperature must the gas be raised, assuming the initial temperature is T?

• A. 3T
• B. 9T
• C. T/3
• D. 6T

Answer: (B)

Verified step by step guidance

1. Recall the formula for the root mean square (rms) speed of gas molecules: \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(R\) is the gas constant, \(T\) is the temperature in kelvin, and \(M\) is the molar mass of the gas.
2. Note that the rms speed is proportional to the square root of the temperature, so we can write \(v_{rms} \propto \sqrt{T}\).
3. If the rms speed is to be tripled, set up the relationship: \$3 v_{rms, initial} = v_{rms, final}\(, which implies \)3 \sqrt{T} = \sqrt{T_{final}}$.
4. Square both sides of the equation to eliminate the square roots: \((3 \sqrt{T})^2 = (\sqrt{T_{final}})^2\), which simplifies to \$9T = T_{final}$.
5. Conclude that the final temperature must be 9 times the initial temperature, so \(T_{final} = 9T\).