The specific heat c of a solid at low temperature shows temperature dependence according to the relation `c = D T^3` where D is a constant and T is the temperature in kelvin. A piece of this solid of mass m kg is taken and its temperature is raised from 20 K to 30 K. The amount of the heat required in the process in energy units is

Options

A. `5xx10^4` Dm
B. `(33//4)xx10^4` Dm
C. `(65//4)xx10^4` Dm
D. `(5//4)xx10^4` Dm

Explanation

To solve the problem, we need to calculate the amount of heat required to raise the temperature of a solid from 20 K to 30 K, given that its specific heat capacity \( c \) is temperature-dependent and given by the equation \( c = D T^3 \).

### Step-by-Step Solution:

1. **Identify the Variables:**
- Mass of the solid: \( m \) (in kg)
- Initial temperature: \( T_1 = 20 \, \text{K} \)
- Final temperature: \( T_2 = 30 \, \text{K} \)
- Specific heat capacity: \( c = D T^3 \)

2. **Understand the Heat Transfer Formula:**
The heat \( Q \) required to change the temperature of a substance is given by:
\[ Q = m \cdot c \cdot \Delta T \]
where \( \Delta T = T_2 - T_1 \).

3. **Substitute the Specific Heat Capacity:**
Since \( c \) is dependent on temperature, we need to express \( Q \) in terms of an integral:
\[ Q = m \int_{T_1}^{T_2} c \, dT = m \int_{T_1}^{T_2} D T^3 \, dT \]

4. **Set Up the Integral:**
Substitute \( c \) into the integral:
\[ Q = m D \int_{20}^{30} T^3 \, dT \]

5. **Calculate the Integral:**
The integral of \( T^3 \) is:
\[ \int T^3 \, dT = \frac{T^4}{4} \]
Therefore, we evaluate:
\[ Q = m D \left[ \frac{T^4}{4} \right]_{20}^{30} \]

6. **Evaluate the Limits:**
Plugging in the limits:
\[ Q = m D \left( \frac{30^4}{4} - \frac{20^4}{4} \right) \]

7. **Calculate \( 30^4 \) and \( 20^4 \):**
- \( 30^4 = 810000 \)
- \( 20^4 = 160000 \)

8. **Substitute Back:**
\[ Q = m D \left( \frac{810000 - 160000}{4} \right) = m D \left( \frac{650000}{4} \right) \]

9. **Simplify:**
\[ Q = \frac{650000}{4} m D = 162500 m D \]

### Final Answer:
The amount of heat required in energy units is:
\[ Q = 162500 \, m D \]