Thermal properties of materials
According to the Grüneisen relation, the volume expansion coefficient of a solid is roughly proportional to its constant-volume heat capacity. Which statement correctly describes the linear expansion coefficient of an insulator at low temperatures near absolute zero?
- AIt keeps nearly the same value as at room temperature even as the temperature falls
- BIt grows as the temperature falls and is largest at absolute zero
- CIt decreases with temperature just as the heat capacity does, approaching 0 at absolute zero
Show answer and explanation
Answer: C. It decreases with temperature just as the heat capacity does, approaching 0 at absolute zero
By the Grüneisen relation β = γC_v/(BV) (β is the volume expansion coefficient, γ the Grüneisen parameter, C_v the constant-volume molar heat capacity, B the bulk modulus and V the molar volume), the thermal expansion coefficient follows nearly the same temperature dependence as the heat capacity. At temperatures well below the Debye temperature the heat capacity shrinks in proportion to T³, so the linear expansion coefficient of an insulator also approaches 0 in proportion to T³. Thermal expansion arises from the anharmonicity of vibrations, so at very low temperatures, where vibrations are hardly excited, expansion also hardly occurs. Using the room-temperature value all the way down to very low temperatures greatly overestimates the shrinkage on cooling. Above the Debye temperature the heat capacity is nearly constant, so the linear expansion coefficient is also nearly constant.