Kinetic theory link:\nFor molecular speeds in a gas, the characteristic speed (e.g., root-mean-square speed) varies with absolute temperature T as __________.

Introduction / Context:
The kinetic theory of gases relates molecular speed distributions to temperature and molecular mass. The root-mean-square (rms) speed, most probable speed, and average speed all scale with the square root of temperature for a given gas.

Given Data / Assumptions:

• Ideal-gas behavior; negligible intermolecular potential effects for the scaling argument.
• Fixed molar mass M for the gas under consideration.

Concept / Approach:
The rms speed is u_rms = √(3 R T / M) for a gas of molar mass M. Both the average speed and the most probable speed have similar √T dependence with different constants (u_avg = √(8 R T / (π M)), u_mp = √(2 R T / M)). Therefore, any representative molecular speed scales as the square root of absolute temperature, not linearly with T.

Step-by-Step Solution:
Recall u_rms = √(3 R T / M).Hold M and R constant for a given gas.Therefore, u_rms ∝ √T.

Verification / Alternative check:
Plotting measured speed versus temperature on log–log axes gives a slope of 0.5, confirming the square-root dependence.

Why Other Options Are Wrong:
Proportional to T or T^2: would vastly overpredict speed increases with temperature.Inversely proportional to T: contradicts the fundamental kinetic theory relation.

Common Pitfalls:
Confusing pressure dependence with temperature scaling; assuming linearity out of habit; ignoring molecular mass in the complete expression.

Final Answer:
directly proportional to √T