Which statement best captures the definition of thermal conductivity k for a homogeneous material?
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AHeat per second through 1 cm² area and 1 cm thickness for 1°C temperature difference
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BHeat per second across 1 cm of material with 1°C temperature difference
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CHeat conducted per unit time across unit area through unit thickness when the temperature difference across faces is unity
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DAll of the above are equally precise and general definitions
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EHeat required to raise temperature by 1°C per unit mass
Answer
Correct Answer: Heat conducted per unit time across unit area through unit thickness when the temperature difference across faces is unity
Explanation
Introduction / Context:Thermal conductivity k quantifies a material’s ability to conduct heat. It appears in Fourier’s law of conduction and is essential for sizing insulation, walls, and heat exchangers. A clear, general definition avoids ambiguity in units and geometry.
Given Data / Assumptions:
- Fourier’s law (one-dimensional steady conduction): q = −k * A * (dT/dx).
- Unit area and unit thickness provide a normalized definition.
Concept / Approach:The most general definition: k is the heat flow rate per unit area for a unit temperature gradient. Equivalently, it is the heat flow per unit time across unit area with unit thickness when the faces differ by 1°C (or 1 K). This avoids specific centimeter-based units and is independent of particular geometry choices beyond “unit.”
Step-by-Step Solution:
1) Start from q/A = −k * ΔT/Δx.2) Set Δx = 1 and ΔT = 1 → q/A = k.3) Therefore, k equals the heat flow per unit area when unit thickness and unit temperature difference are imposed.Verification / Alternative check:Dimensional analysis yields k with units W/(m·K) in SI, consistent with the chosen statement.
Why Other Options Are Wrong:
- (a) and (b) use centimeter-based special cases and are less general.
- (d) is too broad; not all are equally precise.
- (e) defines specific heat, not conductivity.
Common Pitfalls:Mixing up specific heat (energy to raise temperature) with conductivity (rate of heat transfer through a material).
Final Answer:Heat conducted per unit time across unit area through unit thickness when the temperature difference across faces is unity