According to Debye`s law the specific heat at extremely low temperatures value with temperature T as

Options

A. `T`
B. `T^2`
C. `T^3`
D. `T^4`

Answer

To solve the question regarding Debye's law and its relation to specific heat at extremely low temperatures, we can follow these steps:

### Step 1: Understand Debye's Law
Debye's law states that the specific heat capacity (C) of a solid at low temperatures is proportional to the cube of the temperature (T). Mathematically, this can be expressed as:
\[ C \propto T^3 \]

### Step 2: Write the Mathematical Expression
From the proportionality, we can express the specific heat capacity as:
\[ C = kT^3 \]
where \( k \) is a constant that depends on the material.

### Step 3: Analyze the Implications
This relationship implies that as the temperature approaches absolute zero (0 K), the specific heat capacity also approaches zero. This is consistent with the third law of thermodynamics, which states that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero.

### Step 4: Conclusion
Thus, according to Debye's law, the specific heat at extremely low temperatures varies with temperature \( T \) as:
\[ C \approx kT^3 \]