Problem 14 - Chemistry Matter and Change

Question

How much heat is absorbed by a \(2.00 \times 10^{3}-\mathrm{g}\) granite boulder \(\left(\mathrm{c}_{\text { granite }}=0.803 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)\right)\) as its temperature changes from \(10.0^{\circ} \mathrm{C}\) to \(29.0^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The heat absorbed by the granite boulder is 30,514 J.

Step 1: Understand the Formula

To calculate the heat absorbed by the boulder, we use the formula for specific heat transfer: \[ q = m \cdot c \cdot \Delta T \] where \(q\) is the heat absorbed, \(m\) is the mass of the object, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature.

Step 2: Identify the Given Values

We are given:- Mass, \(m = 2.00 \times 10^{3} \text{ g}\)- Specific heat capacity, \(c = 0.803 \text{ J/(g}\cdot\degree\text{C)}\)- Initial temperature, \(T_i = 10.0\degree\text{C}\)- Final temperature, \(T_f = 29.0\degree\text{C}\)The change in temperature, \(\Delta T\), is calculated as \(T_f - T_i\).

Step 3: Calculate the Temperature Change

Calculate the change in temperature:\[ \Delta T = 29.0^{\circ} \text{C} - 10.0^{\circ} \text{C} = 19.0^{\circ} \text{C} \]

Step 4: Calculate the Heat Absorbed

Substitute the known values into the formula for specific heat transfer:\[q = 2.00 \times 10^{3} \text{ g} \times 0.803 \text{ J/(g}\cdot\degree\text{C)} \times 19.0^{\circ} \text{C}\]\[q = 30,514 \text{ J}\]Thus, the heat absorbed by the granite boulder is \(30,514 \text{ J}\).

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept that involves the movement of thermal energy from one object or substance to another. It plays a crucial role in understanding how objects warm up or cool down. In this exercise, the heat transfer occurs as energy is absorbed by the granite boulder, increasing its temperature as it warms up from the environment.
When two objects at different temperatures come into contact, heat energy will naturally flow from the hotter object to the cooler object. This process continues until thermal equilibrium is reached, meaning both objects reach the same temperature.

• Conduction: Heat transfer through direct contact, like when you touch a hot surface.
• Convection: Transfer through a fluid (liquid or gas) medium, like boiling water.
• Radiation: Transfer without a medium, such as the sun's rays warming your skin.

The rate of heat transfer can be influenced by several factors, including the difference in temperature, the specific heat capacities of the materials involved, and the size and shape of the objects.

Temperature Change

Temperature change is at the heart of understanding how much energy is being transferred into or out of a substance. It's simply the difference between the final temperature and the initial temperature. In mathematical terms, this is expressed as \( \Delta T = T_f - T_i \). For our granite boulder, the initial and final temperatures are \(10.0\degree C\) and \(29.0\degree C\) respectively, making the temperature change \(19.0\degree C\).
A larger temperature change generally indicates a larger amount of energy transfer. This tells us how effective the heat transfer has been for the substance in question.
It's important to remember that temperature is a measure of the average kinetic energy of particles in a substance. When heat is added, particles move faster, increasing the temperature. Conversely, when heat is removed, particles slow down, and the temperature drops.

Energy Calculations

Calculating the energy involved in heat transfer is fundamental to understanding specific heat capacity. Specific heat capacity is the amount of heat energy needed to raise the temperature of one gram of a substance by one degree Celsius. It determines how a material will respond to heat input.
In our exercise, the formula used for energy calculations is:
\[ q = m \cdot c \cdot \Delta T \]

• \(q\): Amount of heat energy absorbed or released (in Joules).
• \(m\): Mass of the substance (in grams).
• \(c\): Specific heat capacity of the substance (in J/(g\cdot\degree C)).
• \(\Delta T\): Change in temperature (in \degree C).

By plugging in the given values into the formula, \(2.00 \times 10^{3} \text{ g}\) for mass, \(0.803 \text{ J/(g}\cdot\degree C)\) for specific heat, and \(19.0\degree C\) for temperature change, we calculate \(q\) as \(30,514 \text{ J}\). This is the total energy absorbed by the boulder, showing how much heat was needed to achieve the temperature increase.