The specific heat of a metal at low temperature varies according to `S= (4//5)T^(3)` where `T` is the absolute temperature. Find the heat energy needed to raise unit mass of the metal from T = 1 K ` to `T= 2K`.

Explanation

To find the heat energy needed to raise the unit mass of the metal from T = 1 K to T = 2 K, we will use the given specific heat capacity formula and perform integration. Here’s a step-by-step solution:

### Step 1: Understand the specific heat formula
The specific heat \( S \) of the metal is given by:
\[ S = \frac{4}{5} T^3 \]
where \( T \) is the absolute temperature in Kelvin.

### Step 2: Set up the heat energy equation
The heat energy \( Q \) required to raise the temperature of a unit mass (1 kg) of the metal from temperature \( T_1 \) to \( T_2 \) can be expressed as:
\[ Q = \int_{T_1}^{T_2} S \, dT \]
Since we are raising the temperature from \( T_1 = 1 \) K to \( T_2 = 2 \) K, we can substitute these values into the equation.

### Step 3: Substitute the specific heat into the integral
Substituting \( S \) into the integral, we have:
\[ Q = \int_{1}^{2} \frac{4}{5} T^3 \, dT \]

### Step 4: Factor out constants from the integral
The constant \( \frac{4}{5} \) can be factored out of the integral:
\[ Q = \frac{4}{5} \int_{1}^{2} T^3 \, dT \]

### Step 5: Perform the integration
Now, we need to integrate \( T^3 \):
\[ \int T^3 \, dT = \frac{T^4}{4} \]
Thus, we can write:
\[ Q = \frac{4}{5} \left[ \frac{T^4}{4} \right]_{1}^{2} \]

### Step 6: Evaluate the definite integral
Now we evaluate the integral from 1 to 2:
\[ Q = \frac{4}{5} \left( \frac{2^4}{4} - \frac{1^4}{4} \right) \]
Calculating the values:
\[ Q = \frac{4}{5} \left( \frac{16}{4} - \frac{1}{4} \right) = \frac{4}{5} \left( 4 - \frac{1}{4} \right) = \frac{4}{5} \left( \frac{16}{4} - \frac{1}{4} \right) = \frac{4}{5} \left( \frac{15}{4} \right) \]

### Step 7: Simplify the expression
Now, simplifying:
\[ Q = \frac{4 \times 15}{5 \times 4} = \frac{60}{20} = 3 \]

### Final Answer
Thus, the heat energy needed to raise the unit mass of the metal from 1 K to 2 K is:
\[ Q = 3 \text{ joules} \]