Problem 21 - Chemical Principles: The Quest for Insight

Question

A calorimeter was calibrated with an electric heater, which was used to supply \(22.5 \mathrm{~kJ}\) of energy to the calorimeter; the heat increased the temperature of the calorimeter and its water bath from \(22.45^{\circ} \mathrm{C}\) to \(23.97^{\circ} \mathrm{C}\). What is the heat capacity of the calorimeter?

Step-by-Step Solution

Answer

The heat capacity of the calorimeter is calculated by dividing the supplied heat energy by the temperature change. Using the formula C = Q / ∆T, where Q = 22.5 kJ and ∆T = 1.52°C, the heat capacity is 14.8 kJ/°C.

Step 1: Calculate the temperature change

Find the change in temperature (∆T) of the water by subtracting the initial temperature from the final temperature.∆T = T_final - T_initial = 23.97°C - 22.45°C

Step 2: Apply the heat capacity formula

The heat capacity (C) is the amount of heat energy required to raise the temperature of an object by 1°C. It can be calculated using the formula Q = C * ∆T, where Q is the heat energy supplied. Rearrange this formula to solve for C.C = Q / ∆T

Step 3: Plug in the values and compute the heat capacity

Substitute the values for Q (22.5 kJ) and ∆T (the result from Step 1) into the formula.C = 22.5 kJ / ∆T

Step 4: Convert the heat energy to the same units as heat capacity

Ensure that the heat energy, Q, is in the same units as the heat capacity, which is typically J/°C or kJ/°C. In this case, Q is already given in kJ, so no conversion is necessary.

Step 5: Calculate the final value for the heat capacity

After plugging in the values into the formula from Step 3, perform the division to find the heat capacity of the calorimeter.C = 22.5 kJ / (23.97°C - 22.45°C)

Key Concepts

Temperature Change

Understanding temperature change is crucial when studying heat transfer in thermodynamics. Temperature change, symbolized as \( \Delta T \), represents the difference between the final and initial temperatures of a substance or system. To compute this change, you simply subtract the initial temperature (\( T_{initial} \) from the final temperature (\( T_{final} \):\[ \Delta T = T_{final} - T_{initial} \]In practical terms, if you have a glass of water initially at room temperature, say \( 22^\circ\text{C} \) and you heat it until it reaches \( 30^\circ\text{C} \) the temperature change is \( 8^\circ\text{C} \). Understanding how temperature change is calculated is an integral part of solving problems in calorimetry and thermodynamics.

Heat Capacity Formula

Heat capacity is a property that measures the amount of heat needed to change the temperature of an object by a certain amount, usually by one degree Celsius. The formula for calculating heat capacity (\( C \)) is:\[ C = \frac{Q}{\Delta T} \]where \( Q \) is the heat energy added or removed, and \( \Delta T \) is the temperature change of the object. Heat capacity has units of joules per degree Celsius (J/°C) or kilojoules per degree Celsius (kJ/°C). For instance, if a calorimeter absorbs \( 22.5 \text{kJ} \) to raise its temperature from \( 22.45^\circ\text{C} \) to \( 23.97^\circ\text{C} \) as stated in our example, using the aforementioned formula will allow the determination of its heat capacity. It's an essential concept for understanding how different materials respond to heat.

Calorimetry

Calorimetry is an experimental technique used to measure the amount of heat exchanged in chemical reactions or physical changes. A device called a calorimeter is often used in this process. The principle of calorimetry is based on the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another.

Application in an Experiment

Calorimetry experiments involve an isolated system where no heat is lost to the surroundings. By measuring the temperature change within the system, \( \Delta T \), and knowing the amount of energy supplied, one can calculate the heat capacity of the system. The experiment described involves using an electric heater to transfer a known amount of heat energy to the calorimeter, thus allowing for the calculation of its heat capacity. This technique is widely used in various fields of science and engineering to understand heat transfers.

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It involves studying how energy transformations can compel changes in the physical state and temperature of substances.

Laws of Thermodynamics

• The zeroth law states that if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
• The first law, also known as the law of energy conservation, asserts that energy cannot be created or destroyed in an isolated system.
• The second law states that entropy, or disorder, within an isolated system always increases over time.
• The third law suggests that as the temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum.

These laws lay the foundation for principles used in calorimetry, such as observing the heat capacity to understand how substances absorb or release heat. By applying the concepts of thermodynamics, scientists and engineers can predict how systems will respond to changes in their environment and design processes that are more efficient and sustainable.