Question

A swimming pool, \(10.0 \mathrm{~m}\) by \(4.0 \mathrm{~m}\), is filled with water to a depth of \(3.0 \mathrm{~m}\) at a temperature of \(20.2^{\circ} \mathrm{C}\). How much energy is required to raise the temperature of the water to \(24.6^{\circ} \mathrm{C} ?\)

Short answer

= \(10.0\mathrm{~m} * 4.0\mathrm{~m} * 3.0\mathrm{~m} = 120\mathrm{~m^3}\) #tag_title# Step 2: Calculate the mass of the water #tag_content# To calculate the mass of the water, we need to know the volume (V) and the density (\(\rho\)) of the water. The density of water is approximately \(1000\mathrm{~kg/m^3}\). The mass (m) of the water can be calculated as: m = V * \(\rho\) = \(120\mathrm{~m^3} * 1000\mathrm{~kg/m^3} = 120,000\mathrm{~kg}\) #tag_title# Step 3: Apply the formula for calculating the energy required to heat the water #tag_content# To calculate the energy (Q) required to raise the temperature of the water, we need to know the mass (m) of the water, the specific heat capacity (c) of the water, and the temperature change (\(\Delta T\)). The specific heat capacity of water is approximately \(4.18\mathrm{~kJ/kg\cdot K}\). The temperature change can be calculated as the final temperature minus the initial temperature: \(\Delta T = 24.6^{\circ}\mathrm{C} -20.2^{\circ}\mathrm{C}= 4.4\mathrm{^{\circ}C}\). The energy required (Q) can be calculated as: Q = m * c * \(\Delta T\) = \(120,000\mathrm{~kg} * 4.18\mathrm{~kJ/kg\cdot K} * 4.4\mathrm{~K} = 2,203,776\mathrm{~kJ}\) Thus, the energy required to raise the temperature of the water to \(24.6^{\circ}\mathrm{C}\) is \(2,203,776\mathrm{~kJ}\).

Step-by-step solution

Calculate the volume of water in the pool

To calculate the volume of the water in the pool (V), we need to know the length (L), width (W), and depth (D) of the pool. We are given that the pool is \(10.0\mathrm{~m}\) by \(4.0\mathrm{~m}\), and filled with water to a depth of \(3.0\mathrm{~m}\). The volume of water (V) in the pool can be calculated as: V = L * W * D

Key concepts

Specific Heat Capacity

Specific heat capacity is a core concept in the study of calorimetry. It describes how much energy is required to raise the temperature of a certain mass of a substance by one degree Celsius. It's specific to each material due to its unique molecular structure and bonds. For water, which is often used in calorimetry problems like the swimming pool example, the specific heat capacity is commonly known as 4.18 Joules per gram per degree Celsius (J/g°C). This means that it takes 4.18 Joules of energy to raise the temperature of 1 gram of water by 1°C. When dealing with larger volumes of water, like the water in a swimming pool, it's often more convenient to use the specific heat capacity in terms of energy per kilogram or even per liter if the volume is known. Understanding this property is crucial for calculating how much energy is needed for a given temperature change, as we'll explore further.

Energy Calculation

In calorimetry, energy calculation involves determining the amount of energy required to achieve a temperature change in a substance. The formula used is:\[Q = m \cdot c \cdot \Delta T\]where:

• \(Q\) is the heat energy in Joules,
• \(m\) is the mass of the substance (here it's water),
• \(c\) is the specific heat capacity of the substance, and
• \(\Delta T\) is the change in temperature.

For the swimming pool exercise, once you know the volume and specific heat capacity, you can calculate \(Q\) by determining the total mass of the water (using its density, typically 1000 kg/m³ for water) and using the specific heat capacity of water. This will provide the total energy needed to heat the pool's water to the desired temperature.

Temperature Change

Temperature change \(\Delta T\) is simply the difference between the initial and final temperatures. For the problem involving the swimming pool, you need to find how much the temperature of the water needs to be increased.To calculate \(\Delta T\), you subtract the initial temperature from the final temperature:\[\Delta T = T_{final} - T_{initial}\] For the pool exercise, \(T_{initial}\) is given as 20.2°C and \(T_{final}\) as 24.6°C. Thus:\[\Delta T = 24.6^{\circ}C - 20.2^{\circ}C = 4.4^{\circ}C\]Understanding this change is essential in determining how much energy needs to be applied or removed to achieve the desired temperature. Once \(\Delta T\) is known, it can be plugged into the energy calculation formula along with the other known values to find the total energy required.