The specific heat c of a solid at low temperature shows temperature dependence according to the relation `c = DT^(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 terriperature is raised from 20 K to 30 K. The amount of heat required in the process in energy units

Options

A. `5 xx 10^(4) Dm`
B. `(33//4) xx 10^(4) Dm`
C. `(65//4) xx 10^(4) Dm`
D. `(5//4) xx 10^(4) Dm`

Answer

c

Explanation

The amount of heat supplied or gained by a body is directly proportional to mass of body and amount of change in temperature , i.e., `Delta Q = mc Delta T`
Therefore , amount of heat supplied or gaind `Q = int Delta Q = underset(T_(1)) overset (T_(2)) int mc Delta T `
Here, `c = DT^(3) , T_(1) = 20 K " and " T_(2) = 30 K`
` :. Q = underset(20)overset(30) int mDT^(3) dT = mD underset(20)overset(30) int T^(3) dT = mD [T^(4)/4]_(20)^(30)`
` = (mD)/4 [(30)^(4) - (20)^(4)] = (mD)/4 xx 10^(4) [81 - 16]`
` = (65/4) xx 10^(4) Dm`