Kinetic theory scaling:\nAccording to kinetic theory, the thermal conductivity of a monatomic ideal gas varies with temperature approximately as

Introduction / Context:
For dilute monatomic gases, kinetic theory links transport properties (viscosity, thermal conductivity, and diffusivity) to molecular speed and mean free path. Recognizing the temperature scaling helps estimate heat transfer in gas streams without full property tables.

Given Data / Assumptions:

• Dilute, monatomic, ideal gas.
• Hard-sphere or similar collision model; constant collision integral over modest T ranges.
• Constant pressure comparisons (mean free path varies with T).

Concept / Approach:
Thermal conductivity k for a gas scales roughly as k ~ (1/3) * c_v * ρ * v_bar * λ, where v_bar ~ T^0.5 is mean molecular speed and λ is mean free path. At constant pressure, λ ~ T / pσ ~ T (with weak T-dependence in σ). Combine to see an overall sub-linear scaling; with more complete treatment, the net dependence simplifies to approximately T^0.5 for monatomic gases over common ranges.

Step-by-Step Solution:
Mean speed: v_bar ∝ T^0.5.Heat capacity per molecule is constant (monatomic).Accounting for λ and ρ variations at constant pressure yields an overall k ∝ T^0.5 trend.

Verification / Alternative check:
Empirical correlations (e.g., Sutherland-type) show thermal conductivity and viscosity increasing approximately with T^0.5 to T^0.7 over moderate ranges.

Why Other Options Are Wrong:
Linear or stronger powers (T, T^1.5, T^2) overpredict; “independent of T” contradicts both theory and measurements.

Common Pitfalls:
Confusing constant-volume vs constant-pressure effects on ρ and λ; using high-T collision integrals without care.

Final Answer:
proportional to T^0.5