Problem 75 - Chemistry Matter and Change

Question

Swimming Pool A swimming pool measuring 20.0 \(\mathrm{m} \times 12.5 \mathrm{m}\) is filled with water to a depth of 3.75 \(\mathrm{m} .\) If the initial temperature is \(18.4^{\circ} \mathrm{C}\) , how much heat must be added to the water to raise its temperature to \(29.0^{\circ} \mathrm{C} ?\) Assume that the density of water is 1.000 \(\mathrm{g} / \mathrm{mL}\) .

Step-by-Step Solution

Answer

The pool requires approximately 41.75 GJ of heat to raise the temperature to 29.0°C.

Step 1: Find the Pool Volume

First, calculate the volume of the swimming pool using the formula for volume: \( \text{Volume} = \text{length} \times \text{width} \times \text{depth} \). Using the given measurements: \( \text{Volume} = 20.0 \text{ m} \times 12.5 \text{ m} \times 3.75 \text{ m} = 937.5 \text{ m}^3 \).

Step 2: Convert Volume to Mass

Since 1 cubic meter is 1,000,000 milliliters and given the density of water is 1 g/mL, the mass of the water is equal to the volume in cubic meters converted to grams: \(937.5 \text{ m}^3 \times 1,000,000 \text{ mL/m}^3 = 937,500,000 \text{ g}\). Convert this to kilograms by dividing by 1,000: \(937,500,000 \text{ g} \div 1,000 = 937,500 \text{ kg}\).

Step 3: Determine Heat Required

Use the formula for heat energy: \( q = mc\Delta T \), where \(m\) is the mass of the water, \(c\) is the specific heat capacity of water \((4.186 \text{ J/g}^\circ\text{C})\), and \(\Delta T\) is the change in temperature. The change in temperature is \(29.0^\circ C - 18.4^\circ C = 10.6^\circ C \).

Step 4: Calculate the Heat Energy

Substitute the values into the formula: \( q = 937,500,000 \text{ g} \times 4.186 \text{ J/g}^\circ\text{C} \times 10.6^\circ C \). Calculate the result: \( q = 41,753,475,000 \text{ J} \) or \(41.75 \text{ GJ} \).

Step 5: Final Result

Summarize the final heat energy required, which is approximately \(41.75 \text{ Gigajoules} \).

Key Concepts

Specific Heat Capacity

Specific heat capacity is a fundamental concept in thermodynamics that tells us how much heat is required to change the temperature of a given mass of substance by one degree Celsius. For water, this value is

• 4.186 J/g°C

This means it takes 4.186 Joules of energy to raise the temperature of one gram of water by one degree Celsius.
Knowing the specific heat capacity of water is especially useful because water is a common substance we deal with in everyday problems, like heating swimming pools or cooking.
When calculating heat transfer, this constant enables us to determine the total energy required to reach a desired temperature change when we also consider the mass of the water and the actual change in temperature.

Mass and Volume Calculations

In our swimming pool problem, calculating the mass and volume are key steps. First, we find the pool's volume, which tells us how much space the water occupies. The formula to calculate the volume is:

• Volume = length × width × depth

Using the dimensions provided for the pool, the volume calculates to 937.5 m³.
Next, we need the mass, since heat calculations need mass in addition to specific heat capacity. Fortunately, water has a known density of 1 g/mL or 1,000 kg/m³, making the conversion straightforward. The mass of the water in our pool is:

• Volume × 1,000 kg/m³ = 937,500 kg

This conversion is vital as it bridges the pool's physical dimensions to the calculations needed for determining the heat required to change its temperature.

Temperature Change Analysis

Temperature change analysis is about understanding how much a substance's temperature shifts concerning heat added or removed. We use the formula:

• \[ q = mc\Delta T \]

where \( q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.
For our swimming pool exercise, the temperature change \( \Delta T \) is the difference between the final desired temperature \( 29.0^{\circ} \text{C} \) and initial temperature \( 18.4^{\circ} \text{C} \). This gives us a \( \Delta T \) of 10.6°C.
By inputting this \( \Delta T \) along with the pool's mass and water's specific heat capacity into the formula, we calculate the crucial piece of information: the energy required to achieve this temperature change. The result of our calculation is a significant 41.75 GJ (Gigajoules), illustrating the substantial energy needed for warming a large amount of water.