The molar heat capacity of rock salt at low temperatures...

The molar heat capacity of rock salt at low temperatures varies with temperature according to Debye's $T^{3}$ law". Thus, $C=k \frac{T^{3}}{\theta^{3}}$ where, $k=1940 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \theta=281 \mathrm{~K}$. How much heat is required to raise the temperature of 2 moles of rock salt from $10 \mathrm{~K}$ to $50 \mathrm{~K}$ ? (a) $800 \mathrm{~J}$ (b) $373 \mathrm{~J}$ (c) $273 \mathrm{~J}$ (d) None of these

The molar heat capacity of rock salt at low temperatures varies with temperature according to Debye's $T^{3}$ law". Thus, $C=k \frac{T^{3}}{\theta^{3}}$ where, $k=1940 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \theta=281 \mathrm{~K}$.
How much heat is required to raise the temperature of 2 moles of rock salt from $10 \mathrm{~K}$ to $50 \mathrm{~K}$ ?
(a) $800 \mathrm{~J}$
(b) $373 \mathrm{~J}$
(c) $273 \mathrm{~J}$
(d) None of these

Key Concepts

Temperature-dependent Heat Capacity

In many physical systems, the heat capacity is not constant but depends on temperature. When heat capacity varies with temperature, as it does under Debye’s T³ law, the amount of energy required to change temperature must be calculated by integrating the temperature-dependent heat capacity over the range of temperatures.

Debye T³ Law

This law describes how the heat capacity of a crystalline solid varies as the cube of the temperature at low temperatures. It explains that the number of available vibrational modes (phonons) in the solid increases as T³, leading to a molar heat capacity proportional to T³ in the low-temperature limit.

Integration to Determine Heat Transfer

To find the total heat required to change the temperature of a substance when its heat capacity is temperature-dependent, the integration of the heat capacity function with respect to temperature is performed. This integral represents the cumulative energy absorbed as the temperature increases from the initial to the final value.

Molar Quantities in Thermodynamics

Thermodynamic calculations often involve molar quantities because material properties are given per mole. The total heat required for a process is obtained by multiplying the heat calculated for one mole by the number of moles involved, which is essential for scaling the calculation to the actual amount of substance.

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