Question

The specific heat of most solids is nearly constant over a wide temperature range. Not so for diamond. Between 200 K and 600 K,

the specific heat of diamond is reasonably well described by c = 2.8T - 350 J/kg K, where T is in K. For gemstone diamonds, 1 carat = 200 mg. How much heat energy is needed to raise the temperature of a 3.5 carat diamond from -50°C to 250°C?

Short answer

The amount of heat is 146J

Step-by-step solution

Definition of temperature

Temperature is measured by coldness and hotness which are accounted for with different scales.

Finding the amount of heat 

The temperature difference belongs from 223k to 523K and $m=7\times {10}^{-4}Kg$

According to the definition of the specific heat c, the temperature raise from 2.5T to 350

Then

localid="1649151436820" $$\begin{gathered} \frac{Q}{m}={\int }_{T_1}^{T_2}c\left(T\right)dT={\int }_{T_1}^{T_2}(2.5T-350)dT={(1.4T-350T)}_{223}^{523} \\ \frac{Q}{m}=2.08\times {10}^{5}J\times Kg \end{gathered}$$

Now

$$\begin{gathered} Q=m{\int }_{T_1}^{T_2}c\left(T\right)dT=2.08\times {10}^{5}J\times Kg\times 7\times {10}^{-4} \\ =146J \end{gathered}$$