Select the incorrect statement regarding heat capacities of elements and mixtures.

Difficulty: Medium

Atomic heat capacities of crystalline solid elements are nearly constant and about 6.2 kcal per kg-atom according to the law of Petit and Dulong.
Atomic heat capacities of solid elements decrease greatly with decreasing temperature, approaching zero at absolute zero when crystalline.
Generally, compounds have lower heat capacities in the liquid state than in the solid state.
Heat capacity of a heterogeneous mixture is additive, but upon forming solutions this additivity may no longer hold.
At room temperature many elements have atomic heat capacities close to 3R on a molar basis.

Correct Answer: Generally, compounds have lower heat capacities in the liquid state than in the solid state.

Explanation:

Introduction / Context:
Thermal properties guide design of heating, cooling, and energy storage operations. Recognizing typical trends—like the Dulong–Petit law and the behavior of heat capacity with temperature and phase—prevents serious design miscalculations.

Given Data / Assumptions:

Dulong–Petit (room temperature) trend: Cp,m ≈ 3R for many crystalline elements.
Debye model: heat capacities drop toward zero as T → 0 K.
For most substances, Cp(liquid) > Cp(solid) near the melting point.

Concept / Approach:
Evaluate each statement against known thermodynamic behavior. Look particularly for a reversal of the typical phase trend between solid and liquid heat capacities.

Step-by-Step Solution:

(a) Interpretation: 6.2 kcal/kg-atom·K equals ~6.2 cal/g-atom·K, consistent with 3R ≈ 5.96 cal/mol·K → broadly correct.(b) Matches Debye behavior as T decreases.(c) Claims Cp(liquid) < Cp(solid) in general—this is contrary to common observation; liquids usually have higher Cp.(d) Additivity holds for simple mixtures by mass-fraction weighting; upon solution formation, excess heat capacities appear—statement is correct.(e) Restates Dulong–Petit in molar form: acceptable.

Verification / Alternative check:
Data tables show Cp(H2O, liquid) > Cp(ice) near 0–25 °C; many organics also show Cp(liquid) > Cp(solid) near fusion temperatures.

Why Other Options Are Wrong:

Only (c) contradicts standard physical trends; others are consistent with theory or empirical data.

Common Pitfalls:
Confusing caloric units (cal vs kcal) per gram-atom vs kg-atom; ignoring temperature dependence and phase effects.

Final Answer:
Generally, compounds have lower heat capacities in the liquid state than in the solid state.

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Select the incorrect statement regarding heat capacities of elements and mixtures.

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