Question

A \(35.2 \mathrm{g}\) sample of metal \(\mathrm{Z}\) requires \(1251 \mathrm{J}\) of energy to heat the sample by \(25.0^{\circ} \mathrm{C}\). Calculate the specific heat capacity of metal Z.

Short answer

The specific heat capacity of metal Z is approximately 1.418 J/g°C.

Step-by-step solution

List the given values

We have the following information: - Mass of metal Z (m): 35.2 g - Energy required to heat the sample (q): 1251 J - Change in temperature (ΔT): 25.0°C

Rearrange the formula to solve for specific heat capacity (c)

We need to find the value of c. Rearrange the formula: \(c = \frac{q}{mΔT}\)

Substitute the given values to calculate the specific heat capacity

Now, substitute the given values into the formula: \(c = \frac{1251 \ \text{J}}{(35.2 \ \text{g})(25.0 \ \text{°C})}\)

Calculate the specific heat capacity of metal Z

By performing the calculation: \(c \approx \frac{1251}{882} = 1.418 \ \text{J/g} \cdot \text{°C}\) So, the specific heat capacity of metal Z is approximately 1.418 J/g°C.

Key concepts

Thermal Energy

Thermal energy refers to the energy that comes from the temperature of matter. It is a form of kinetic energy due to the movement of particles within an object. The faster these particles move, the more thermal energy is produced. When we heat an object, we increase its thermal energy.

In the context of the exercise, thermal energy is the energy required to increase the temperature of the metal sample. It is measured in joules (J), which is the standard unit of energy in the International System of Units (SI).

Understanding thermal energy helps us comprehend how much energy we need to supply to a material to achieve a desired increase in temperature. It also illustrates the principle that different materials require different amounts of energy to undergo the same temperature change.

Specific Heat Formula

The specific heat formula is a mathematical expression used to calculate the specific heat capacity of a substance. The formula is given by:

\[ c = \frac{q}{m \cdot \Delta T} \]

where:

• \(c\) is the specific heat capacity
• \(q\) is the thermal energy absorbed or released
• \(m\) is the mass of the substance
• \(\Delta T\) is the change in temperature

This formula allows you to find the specific heat capacity by rearranging it to solve for \(c\). The specific heat capacity is a property that is unique to each substance, indicating how much energy is needed to raise the temperature of one gram of that substance by one degree Celsius.

Using the specific heat formula, you can understand why different substances heat up or cool down at different rates. For example, water has a high specific heat capacity, meaning it takes more energy to change its temperature compared to many other substances.

Temperature Change Calculation

Calculating temperature change is a crucial step in determining how much energy is exchanged in a physical process. The temperature change (\(\Delta T\)) is the difference between the final temperature and the initial temperature of the substance. It is calculated using the equation:

\[ \Delta T = T_f - T_i \]

where:

• \(T_f\) is the final temperature
• \(T_i\) is the initial temperature

In our exercise, the given temperature change was 25.0°C. This value directly affects the amount of energy required to increase the temperature of the material. A larger temperature change typically requires more energy.

Temperature change calculations are important in various scientific fields, including chemistry and physics, as they help predict how substances behave under different thermal conditions. Understanding how to calculate temperature change is vital when working with the specific heat formula, as it’s an integral part of solving these types of problems.