Problem 156 - Introductory Chemistry Atoms First

Question

When \(10 \mathrm{~kJ}\) of heat energy is added to a beaker containing \(250 \mathrm{~g}\) of water initially at \(23.0{ }^{\circ} \mathrm{C}\), what is the final temperature of the water?

Step-by-Step Solution

Answer

The final temperature of the water when 10 kJ of heat energy is added to the beaker is approximately \(32.57^{\circ}\)C.

Step 1: Identify the given information

We are given the following information: - Heat energy added (Q) = 10 kJ (convert to J by multiplying by 1000, so Q = 10000 J) - Mass of water (m) = 250 g (convert to kg by dividing by 1000, so m = 0.25 kg) - Initial temperature (T1) = 23.0 °C - Specific heat capacity of water (c) = 4.18 kJ/kg·°C = 4180 J/kg·°C We need to find the final temperature (T2).

Step 2: Use the heat transfer formula

The formula for calculating the heat transfer in a substance is: Q = mcΔT where ΔT = T2 - T1

Step 3: Rearrange the formula to solve for the final temperature T2

To find the final temperature T2, we need to rearrange the formula as follows: ΔT = Q / (mc) T2 = T1 + Q / (mc)

Step 4: Plug in the given values and solve for T2

Now plug in the given information into the formula and calculate the final temperature: T2 = 23.0 + (10000 J) / (0.25 kg × 4180 J/kg·°C) T2 = 23.0 + (10000) / (1045) T2 = 23.0 + 9.57 ≈ 32.57 °C The final temperature of the water when 10 kJ of heat energy is added to the beaker is approximately 32.57 °C.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a fundamental concept in calorimetry, crucial for understanding how substances absorb and release heat. It represents the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. In this scenario, the specific heat capacity of water is given as 4180 J/kg·°C.
This value is quite essential because it tells us about the energy absorption capacity of water - a factor which influences its temperature change during heating or cooling. To put it into perspective, water has a relatively high specific heat capacity. This means it requires more energy to change its temperature compared to other substances. As a result, water can store and transport energy efficiently, useful in both natural and industrial processes.

Temperature Change Formula

The temperature change formula is a valuable equation in physics when we want to find out how a substance's temperature changes due to heat exchange. It is expressed as follows:
\( \Delta T = T_2 - T_1 \) where:

• \( \Delta T \) expresses the change in temperature,
• \( T_2 \) is the final temperature,
• \( T_1 \) is the initial temperature.

If we rearrange the formula to find the final temperature \( T_2 \), it becomes:\[ T_2 = T_1 + \frac{Q}{mc} \]In this case, \( Q \) stands for the heat energy added, \( m \) is the mass of the water, and \( c \) is the specific heat capacity. This formula helps us calculate how much a given amount of heat will change the temperature of a particular mass of water.

Final Temperature of Water

Calculating the final temperature of water after adding heat involves applying our understanding of specific heat capacity and the temperature change formula. We know the initial temperature, the amount of heat added, and the specific heat capacity of the water.
By substituting these values into the rearranged formula, we compute the change in temperature and consequently determine the final temperature.
Using the given problem as an example, we start with an initial temperature of 23.0°C and add 10,000 J of heat to 0.25 kg of water. With the specific heat of water at 4180 J/kg·°C, we calculate:\[ T_2 = 23.0 + \frac{10000}{0.25 \times 4180} \]This gives us a final temperature of approximately 32.57°C. Such calculations are common in thermal physics and everyday applications.

Calorimetry

Calorimetry is the study of measuring heat flow in a system, a vital part of thermodynamics. It helps us understand changes in energy when substances absorb or release heat. In essence, calorimetry provides a clear framework for determining temperature changes and energy exchanges.
In practical terms, calorimetry involves using a calorimeter to perform such measurements, but the core principles remain the same in theoretical exercises.

• The amount of heat exchanged, \( Q \), is proportional to the mass \( m \), the specific heat capacity \( c \), and the temperature change \( \Delta T \).
• This relationship is captured in the equation \( Q = mc\Delta T \).

So, whether you're heating water in a classroom activity or analyzing industrial energy efficiency, calorimetry plays a critical role in accurately measuring heat transfer and energy conservation.