EXAMPLE $1\mathrm{.}8$

Temperature$-$Dependent Heat Capacity

At low temperatures, the specific heats of solids are typically proportional to $T^3.$ The first understanding of

this behavior was due to the Dutch physicist Peter Debye, who in $1912,$ treated atomic oscillations with the

quantum theory that Max Planck had recently used for radiation. For instance, a good approximation for the

specific heat of salt, NaCl, is $c=3\mathrm{.}33{10}^4\frac{J}{kg*K}(\frac{T}{321(K)}{)}^3.$ The constant $321K$ is called the Debye

temperature of NaCl,${}_D,$ and the formula works well when Using this formula, how much heat is

required to raise the temperature of $24\mathrm{.}0g$ of NaCl from $5K$ to $15K?$