Heat capacity of solids vs. temperature:\nHow does the atomic (molar) heat capacity of solid elements vary as temperature decreases?

Introduction / Context:
Understanding how heat capacity varies with temperature is central to low-temperature operations, cryogenics, and solid-state process design. Classical Dulong–Petit behavior (approximately constant molar heat capacity near room temperature) breaks down at low temperatures, where quantum effects govern.

Given Data / Assumptions:

• Pure solid elements; temperature decreasing from ambient toward cryogenic conditions.
• No phase transitions in the considered range.

Concept / Approach:
As temperature decreases, vibrational modes in a crystal are progressively “frozen out.” Debye theory predicts C_v ∝ T^3 at sufficiently low temperatures. Consequently, the atomic (molar) heat capacity decreases as temperature falls and tends toward zero in the limit of 0 K (absolute zero). It does not approach zero at 0°C; that option confuses Celsius with Kelvin.

Step-by-Step Solution:
Recall low-temperature limit: C_v ≈ a * T^3 for many solids → decreasing function of T.At intermediate/room temperatures, C_v approaches classical values but still generally drops as T is lowered.Hence, the correct qualitative trend is “decreases.”

Verification / Alternative check:
Handbook curves for copper, aluminum, and silicon all show C_p and C_v diminishing with T, approaching zero as T → 0 K, validating the trend.

Why Other Options Are Wrong:
“Increases” contradicts empirical and theoretical behavior; “unchanged” ignores temperature dependence; “approaches zero at 0°C” is incorrect (should be at 0 K); the “first increases then decreases” pattern is not the general law for elemental solids.

Common Pitfalls:
Confusing Celsius and Kelvin scales; assuming Dulong–Petit holds at all temperatures.

Final Answer:
decreases