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 find the heat energy required to raise the temperature of a mass \( m \) of a solid from \( T = 0 \) K to \( T = 20 \) K, we start with the given relation for specific heat:

\[ S = A T^3 \]

where \( A \) is a constant. 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 \( S \) into the equation, we have:

\[ dq = m (A T^3) dT \]

To find the total heat energy \( Q \) required to raise the temperature from \( 0 \) K to \( 20 \) K, we need to integrate \( dq \):

\[ Q = \int_{0}^{20} m (A T^3) dT \]

Factoring out the constants \( m \) and \( A \):

\[ Q = m A \int_{0}^{20} T^3 dT \]

Now, we can evaluate the integral \( \int T^3 dT \):

\[ \int T^3 dT = \frac{T^4}{4} \]

Now we can substitute the limits of integration from \( 0 \) to \( 20 \):

\[ Q = m A \left[ \frac{T^4}{4} \right]_{0}^{20} \]

Calculating the definite integral:

\[ Q = m A \left( \frac{20^4}{4} - \frac{0^4}{4} \right) \]

Calculating \( 20^4 \):

\[ 20^4 = 160000 \]

So we have:

\[ Q = m A \left( \frac{160000}{4} \right) = m A (40000) \]

Thus, the total heat energy required is:

\[ Q = 40000 m A \]

This can be expressed as:

\[ Q = 4 m A \times 10^4 \]

### 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:

\[ Q = 4 m A \times 10^4 \]