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

The energy required to raise the temperature of the water in the swimming pool from 20.2°C to 24.6°C is approximately 2,210,732,800 joules.

Step-by-step solution

Calculate the volume of water in the pool#

The volume V of the pool can be calculated using the formula V = length × width × height. Using the given dimensions of the pool (10.0 m × 4.0 m × 3.0 m), V = 10.0 m * 4.0 m * 3.0 m = 120.0 m³

Calculate the mass of the water in the pool#

To find the mass of the water, we need to multiply the volume of the water by the density of water (approximately 1,000 kg/m³): m = V × density_water = 120.0 m³ × 1,000 kg/m³ = 120,000 kg

Calculate the change in temperature#

ΔT is the difference between the final temperature and the initial temperature in Celsius or Kelvin: ΔT = T_final - T_initial = 24.6°C - 20.2°C = 4.4°C (or 4.4 K)

Calculate the energy required using the specific heat formula#

Now, we plug in the values for mass, specific heat capacity, and change in temperature into the specific heat formula: Q = mcΔT = 120,000 kg * 4,184 J/(kg·K) * 4.4 K = 2,210,732,800 J The energy required to raise the temperature of the water in the pool from 20.2°C to 24.6°C is approximately 2,210,732,800 joules.

Key concepts

Understanding Specific Heat Capacity

The concept of specific heat capacity is crucial in thermochemistry as it describes how much energy a substance needs to change its temperature by one degree. In simple terms, specific heat capacity tells us how well a material can store thermal energy.
In scientific notation, specific heat capacity is often represented by the symbol \(c\), and measured in units of \(J/(kg\cdot K)\) (joules per kilogram per Kelvin). This unit ensures that we quantify how much energy is required for a given mass to undergo a change in temperature. Understanding this value can help us comprehend why certain materials heat up or cool down faster than others.
For instance, water has a relatively high specific heat capacity of 4,184 \(J/(kg\cdot K)\), meaning it requires a substantial amount of energy to change its temperature compared to many other substances. This characteristic is what allows water to play a significant role in regulating temperatures in natural environments.

Recognizing Temperature Change

Temperature change is another vital concept in determining energy requirements during heating or cooling. In our exercise, calculating the temperature change was straightforward.
We found the difference between the final and initial temperatures: going from 20.2°C to 24.6°C. The change \(\Delta T\) was straightforward to calculate by subtracting the initial from the final, resulting in a 4.4°C increase. It is essential to note that the difference is the same whether we measure it in Celsius or Kelvin. This is because the size of a degree is identical in both scales, ensuring consistent calculations without needing conversions for temperature differences.
Understanding temperature changes is key in calculating how much energy will be required or released during thermal processes.

Carrying Out Energy Calculations

To calculate the energy needed to raise the temperature of a substance, we use the formula: \[ Q = mc\Delta T \]where \(Q\) is the energy in joules, \(m\) is mass in kilograms, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. This formula essentially combines all factors together to determine the total energy exchange.
With the swimming pool exercise, we calculated the mass of water first, which was 120,000 kg. This mass was then used in our energy calculation along with the previously determined specific heat capacity of water (4,184 \(J/(kg\cdot K)\)) and the temperature change (4.4 K).
Plugging these values into the formula gives:\[ Q = 120,000 \: \text{kg} \times 4,184 \: \text{J/(kg}\cdot\text{K)} \times 4.4 \: \text{K} = 2,210,732,800 \: \text{J} \]This total energy result tells us precisely how much energy is needed to bring the pool's water to the desired temperature. Energy calculations like these are foundational in designing heating or cooling systems.