(a) Near room temperature the specific heat capacity of ethanol is \(2.42 \mathrm{~J} \cdot\left({ }^{\circ} \mathrm{C}\right)^{-1} \cdot \mathrm{g}^{-1}\). Calculate the heat that must be removed to reduce the temperature of \(150.0 \mathrm{~g}\) of \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OH}\) from \(50.0^{\circ} \mathrm{C}\) to \(16.6^{\circ} \mathrm{C}\). (b) What mass of copper can be heated from \(15^{\circ} \mathrm{C}\) to \(205^{\circ} \mathrm{C}\) when \(425 \mathrm{~kJ}\) of energy is available?

Understanding heat transfer calculations is crucial when you're tasked with figuring out how much energy is needed to change the temperature of a substance, or in reverse, discovering how a given amount of energy will affect a substance's temperature. These calculations are essential in fields from culinary arts to chemical engineering.

For the heat transfer calculation, the key formula to remember is: \( q = m \times c \times \Delta T \), where \(q\) represents the heat transferred in joules (J), \(m\) is the mass of the substance in grams (g), \(c\) is the specific heat capacity in joules per gram degree Celsius (J/g°C), and \(\Delta T\) is the change in temperature in degrees Celsius (°C).

In the exercise example with ethanol, we observed that a specific amount of heat needed to be removed to cool the ethanol to a lower temperature. When following the steps in the solution, a negative value for \(\Delta T\) is indicative of heat loss, reflecting cooling rather than heating.

It's important to note that the specific heat capacity, \(c\), which is unique for each material, signifies how much energy is required to raise the temperature of one gram of the substance by one degree Celsius. With this concept, we are armed with the knowledge to tackle a variety of practical problems involving heat transfer.

The concept of temperature change is intimately tied with the idea of heat transfer. It's the visible outcome of energy being absorbed or released by a substance. More fundamentally, it's about how particles within a substance respond to the addition or removal of energy.

Different substances respond to heat in varying ways due to their unique specific heat capacities. For instance, water has a high specific heat capacity, meaning it requires a lot of energy to change its temperature. This property makes it excellent for tempering environments.

Within our textbook solution, the temperature change of ethanol, \( \Delta T \), was calculated as the difference between its initial and final temperatures. The result of which was then used in the heat transfer calculation. Similarly, when dealing with the heating of copper, knowing the initial and final temperatures allowed us to determine the energy needed to achieve the desired temperature increase. Remember that a positive \( \Delta T \) indicates heating, while a negative \( \Delta T \) denotes cooling.

The principle of energy conservation lies at the heart of thermodynamics and dictates that energy can’t be created or destroyed, only transformed from one form to another. In the realm of heat transfer, this principle reminds us that the heat energy lost by a substance must be accounted for - perhaps it's being transferred to another substance, or being dispersed into the environment.

In our textbook solutions, we see that by calculating the heat removed from ethanol and the energy required to heat copper, we are applying the conservation of energy. The energy isn't gone; it's simply relocated from or to the environment. Moreover, this principle allows us to solve problems like part (b), where we calculate how much copper can be heated with a given amount of energy. Here, rather than heat being dissipated, it’s being harnessed to increase the copper’s temperature.

Understanding energy conservation helps us not only in isolated calculations but also in designing systems that manage energy efficiently, such as thermal insulation in buildings or cooling systems in electronic devices.