Correct option is A
Given:
- Debye temperature = 150 K.
- The system is a two-dimensional insulator.
- We need to find the ratio of the heat required to raise the temperature from 1 K to 2 K and from 2 K to 3 K.
Solution:
Heat required in a two-dimensional Debye system:
At low temperatures (T << Debye temperature), the heat capacity (C) of a two-dimensional system is proportional to T².
The heat required (Q) to raise the temperature is proportional to the integral of T² over the temperature range:
Heat required Q ∝ ∫ T² dT.Heat required to raise the temperature from 1 K to 2 K (Q1):
Q1 ∝ ∫ T² dT from T = 1 to T = 2.
Evaluate the integral:
Q1 ∝ (2³ / 3) - (1³ / 3) = (8 / 3) - (1 / 3) = 7 / 3.Heat required to raise the temperature from 2 K to 3 K (Q2):
Q2 ∝ ∫ T² dT from T = 2 to T = 3.
Evaluate the integral:
Q2 ∝ (3³ / 3) - (2³ / 3) = (27 / 3) - (8 / 3) = 19 / 3.Ratio of heat required:
The ratio Q1 : Q2 is:
Q1 : Q2 = (7 / 3) : (19 / 3) = 7 : 19.
Conclusion:
The ratio of the heat required to raise the temperature from 1 K to 2 K and from 2 K to 3 K is (a) 7:19.
