How much heat is needed to melt 20.50 kg of silver that is initially at 17 degrees C? The melting...

Question:

How much heat is needed to melt 20.50 kg of silver that is initially at 17 degrees C? The melting point of silver is 961 degrees C, the heat of fusion is 88 kJ/kg, the specific heat is 230 J/kg {eq}\cdot {/eq} degrees C.

Melting Silver:

The energy change that a substance requires to change from one phase to another can be described in two parts. Initially, energy is needed to essentially heat the substance towards a phase change point. When the substance reaches this temperature, it needs a larger amount of energy to be able to change phase.

First, we convert the heat of fusion to J/kg to make it more consistent with everything else:

{eq}H_f = 88\text{ kJ/kg}(\dfrac{1000\text{ J}}{1\text{ kJ}}) \\ H_f = 0.088\text{ J/kg} {/eq}

To melt silver, we need to apply heat to it to elevate its temperature to its melting point, and from the melting point a larger amount of heat to change phase:

{eq}Q = Q_{{\small (17 \rightarrow 961)^\circ\text{C}}} + Q_{{\small \text{melting}}} \\ Q = m c \Delta T + m H_f \\ Q = m (c (961^\circ \text{C} - 17^\circ \text{C}) + H_f) {/eq}

We can simply plug in the given values to solve for the energy needed.

{eq}Q = m (c (961^\circ \text{C} - 17^\circ \text{C}) + H_f) \\ Q = (20.50\text{ kg})((230\ \mathrm{ \frac{J}{kg \cdot ^\circ C}})(961^\circ \text{C} - 17^\circ \text{C}) + (0.088\text{ J/kg})) \\ Q = 4450961.804\text{ J} \\ \boxed{Q = 4.45\text{ MJ}} {/eq}