The specific heat of metals at low temperature is

Options

A. proportional to T
B. proportional to `T^2`
C. proportional to `T^3`
D. Independent of `T`

Answer

To solve the question regarding the specific heat of metals at low temperatures, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Specific Heat**:
Specific heat is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin).

2. **Low Temperature Behavior**:
At low temperatures, the behavior of specific heat for solids, particularly metals, is not constant. Instead, it changes with temperature.

3. **Debye's T³ Law**:
The specific heat of solids at low temperatures can be explained by Debye's T³ law. According to this law, the specific heat (C) at constant volume (V) varies as the cube of the absolute temperature (T) when the temperature is low.

Mathematically, this can be expressed as:
\[ C \propto T^3 \]
This means that as the temperature decreases, the specific heat also decreases and follows a cubic relationship with temperature.

4. **Conclusion**:
Therefore, the specific heat of metals at low temperatures is proportional to \( T^3 \).

5. **Selecting the Correct Option**:
Given the options, the correct answer is the one that states that the specific heat is proportional to \( T^3 \).

### Final Answer:
The specific heat of metals at low temperature is proportional to \( T^3 \) according to Debye's T³ law.

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