A 135 -g sample of a metal requires \(2.50 \mathrm{~kJ}\) to change its temperature from \(19.5^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C}\). What is the specific heat of this metal?

Thermal energy transfer, also known as heat transfer, is the movement of heat from one place to another. Heat can be transferred in three ways: conduction, convection, and radiation. In solids, heat transfer usually occurs through conduction, where heat moves from an area of higher temperature to an area of lower temperature through direct contact between molecules.

In the exercise provided, when a piece of metal absorbs thermal energy, the molecules within the metal start to move faster and the temperature of the metal increases. This transfer of energy is particularly important to note because specific heat capacity is a property that tells us how much energy is needed to raise the temperature of a material.

The transfer of thermal energy is crucial for determining specific heat because it considers the amount of energy needed to raise the temperature of a specific mass of a substance. Thus, understanding how thermal energy transfer works helps us to comprehend why different substances require different amounts of energy to change their temperature by the same amount.

Temperature change calculations involve determining the difference in temperature as a substance absorbs or releases heat. The change in temperature, symbolized as \(\Delta T\), is a crucial component in calculating the specific heat capacity of a substance.

In the exercise, to find \(\Delta T\), we subtract the initial temperature of the metal from its final temperature after it has absorbed heat. Here is the mathematical representation of the temperature change:\[\Delta T = T_{final} - T_{initial}\]
For our specific example, the final temperature is \(100.0^\circ \mathrm{C}\) and the initial temperature is \(19.5^\circ \mathrm{C}\), giving us \(\Delta T = 100.0^\circ \mathrm{C} - 19.5^\circ \mathrm{C} = 80.5^\circ \mathrm{C}\).

Why is Temperature Change Important?

• It helps understand how much a substance responds to a given amount of energy.
• It allows us to apply the specific heat capacity formula correctly to find out the energy required or released during temperature changes.

By accurately calculating temperature change, we can directly link the thermal energy transfer to the resultant increase or decrease in temperature.

The heat capacity formula is critical for calculating the specific heat capacity of substances. The specific heat capacity (\(c\)) is defined as the amount of heat (\(q\)) required to raise the temperature of a unit mass (\(m\)) of a substance by one-degree Celsius (or one Kelvin).

The formula to find specific heat capacity is:\[c = \frac{q}{m \Delta T}\]
Where:\[q\] = heat absorbed or released in joules (J) or kilojoules (kJ)
\[m\] = mass of the substance in grams (g)
\[\Delta T\] = change in temperature, calculated as final temperature minus initial temperature

Using this formula, we can solve the problem provided in the exercise by substituting the appropriate values. First, we converted the given heat from kilojoules to joules for consistency with the mass in grams and temperature change in Celsius. After determining the values of \(m\) and \(\Delta T\), we applied them to the formula.

Importance of the Specific Heat Capacity Formula

• It allows us to calculate the energy required to heat different materials and compare their thermal properties.
• It is essential in various applications, from designing heating systems to understanding environmental physics.

With the formula, we learn not only how substances interact with heat but also how to engineer solutions that require precise temperature control.