The specific heat of many solids at low temperatures varies...

The specific heat of many solids at low temperatures varies with absolute temperature $T$ according to the relation $S=a T^{3}$, where $a$ is a constant. The heat energy required to raise the temperature of a mass $m$ of such a solid from $T=0 \mathrm{~K}$ to $T=20 \mathrm{~K}$ is (a) $4 \times 10^{4} \mathrm{~mA}$ (b) $2 \times 10^{3} \mathrm{~mA}$ (c) $8 \times 10^{6} \mathrm{~mA}$ (d) $2 \times 10^{6} \mathrm{~mA}$

The specific heat of many solids at low temperatures varies with absolute temperature $T$ according to the relation $S=a T^{3}$, where $a$ is a constant. The heat energy required to raise the temperature of a mass $m$ of such a solid from $T=0 \mathrm{~K}$ to $T=20 \mathrm{~K}$ is
(a) $4 \times 10^{4} \mathrm{~mA}$
(b) $2 \times 10^{3} \mathrm{~mA}$
(c) $8 \times 10^{6} \mathrm{~mA}$
(d) $2 \times 10^{6} \mathrm{~mA}$

Key Concepts

Integration of Temperature-Dependent Specific Heat

When the specific heat capacity varies with temperature, calculating the total heat energy required to change the temperature involves integrating the product of the specific heat function and the differential temperature over the desired temperature range. This method ensures that the variable contribution of heat capacity at each temperature is properly accounted for.

Debye T³ Law

The Debye T³ Law is a theoretical model that explains the low-temperature behavior of the specific heat of solids. It predicts that the specific heat varies as the cube of the temperature (T³), due to the quantized nature of lattice vibrations in the crystal structure, and is particularly useful in describing the thermal properties of non-metallic solids at low temperatures.

Specific Heat Capacity

This concept denotes the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree. It is a fundamental property in thermodynamics and can vary with temperature depending on the material and its internal structure.

Temperature Dependence of Specific Heat

In many solids, especially at low temperatures, the specific heat is not constant but instead varies with temperature. This dependency often follows a power law (such as a T³ variation), reflecting how molecular vibrations and energy storage capacities change with thermal energy.

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