The specific heat of many solids at low temperatures varies with absolute temperature T according to the relation `S=AT^(3)`, where A is a constant. The heat energy required to raise the temperature of a mass m of such a solid from T = 0 to T = 20 K is

Options

A. `4xx10^(4)mA`
B. `2xx10^(3)mA`
C. `8xx10^(6)mA`
D. `2xx10^(6)mA`

Answer

To solve the problem of finding the heat energy required to raise the temperature of a mass \( m \) of a solid from \( T = 0 \) K to \( T = 20 \) K, we will follow these steps:

### Step 1: Understand the relationship between specific heat and temperature
The specific heat \( S \) of the solid is given by the equation:
\[ S = A T^3 \]
where \( A \) is a constant and \( T \) is the absolute temperature.

### Step 2: Relate heat energy to specific heat
The heat energy \( dQ \) required to raise the temperature of a mass \( m \) by an infinitesimal amount \( dT \) is given by:
\[ dQ = m S \, dT \]
Substituting the expression for specific heat \( S \):
\[ dQ = m (A T^3) \, dT \]

### Step 3: Set up the integral to find total heat
To find the total heat \( Q \) required to raise the temperature from \( T = 0 \) K to \( T = 20 \) K, we need to integrate \( dQ \):
\[ Q = \int_{0}^{20} m A T^3 \, dT \]

### Step 4: Perform the integration
We can factor out the constants \( m \) and \( A \) from the integral:
\[ Q = m A \int_{0}^{20} T^3 \, dT \]
Now, we compute the integral:
\[ \int T^3 \, dT = \frac{T^4}{4} \]
Evaluating this from \( 0 \) to \( 20 \):
\[ \int_{0}^{20} T^3 \, dT = \left[ \frac{T^4}{4} \right]_{0}^{20} = \frac{20^4}{4} - \frac{0^4}{4} = \frac{160000}{4} = 40000 \]

### Step 5: Substitute back to find \( Q \)
Now substituting back into the expression for \( Q \):
\[ Q = m A \cdot 40000 \]
Thus, the heat energy required to raise the temperature of the mass \( m \) from \( 0 \) K to \( 20 \) K is:
\[ Q = 40000 m A \]

### Final Answer
The heat energy required to raise the temperature of a mass \( m \) of such a solid from \( T = 0 \) K to \( T = 20 \) K is:
\[ Q = 40000 m A \]