Question

Which has more total energy: a cup of boiling water at \(100^{\circ} \mathrm{C}\) or a swimming pool of slightly cooler water at \(90^{\circ} \mathrm{C}\) ?

Short answer

The swimming pool at \(90^{\circ}\mathrm{C}\) has more total energy due to its much larger mass, despite the slightly lower temperature.

Step-by-step solution

Understand the Problem

The problem requires us to compare the total energy in a cup of boiling water at \(100^{\circ} \mathrm{C}\) with a swimming pool at \(90^{\circ} \mathrm{C}\). Total energy refers to the thermal energy contained within the water, linked to both temperature and the quantity of water.

Consider the Quantity of Water

A swimming pool contains significantly more water than a cup. Even though the pool's temperature is lower, the sheer volume can mean more total energy overall.

Recall the Formula for Thermal Energy

The thermal energy \(Q\) stored in an object can be calculated using the formula \(Q = mc\Delta T\), where \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change. Here, \(\Delta T\) is not needed as we're comparing at one time point.

Compare Thermal Energies

Assuming water's specific heat is consistent, the overall energy really depends on mass and temperature. The pool's much larger volume means \(m_{\text{pool}} \gg m_{\text{cup}}\). Combined with a comparatively small temperature difference, the pool will have more total energy due to its volume.

Key concepts

Specific Heat Capacity

Specific heat capacity is a property of a substance that tells us how much energy is needed to raise the temperature of a certain amount of that substance by one degree Celsius. For water, this is relatively high, meaning it takes a lot of energy to change its temperature. When evaluating thermal energy, specific heat capacity plays a crucial role as it determines how energy is stored within the substance.

• In calculations, the specific heat capacity is denoted by the letter \( c \).
• For water, \( c \) is approximately \( 4.18 \text{ J/g}^{ ext{°C}} \). This figure highlights water's ability to "hold" onto energy.

For both a cup and a pool, since the substance is water, the specific heat capacity remains the same. We use this constant to then analyze the impact of mass and temperature on its energy content.

Temperature Difference

Temperature difference, often symbolized as \( \Delta T \), might initially seem like a deciding factor in energy content, yet, in this scenario, the focus is on absolute temperature rather than changes. Here we evaluate the energy based on their current temperatures: \(100^{ ext{°C}}\) for the cup and \(90^{ ext{°C}}\) for the pool. A mere \(10^{ ext{°C}}\) difference can seem significant in small quantities, but when scaled up to larger masses, its impact is diluted.

• When considering \( \Delta T \), it usually refers to a change from one state to another. In this context, it does not play a significant role.
• For the energy calculation, the critical consideration is the temperature of water at a specific moment, influencing how energy is distributed inside it.

Thus, the primary influence in this comparison stems from the huge difference in the mass of the water rather than slight variations in temperature.

Mass of Water

This is a pivotal element in determining thermal energy when comparing two entities containing water. Mass is symbolized by \( m \) and, in our scenario, varies dramatically between a cup of water and a swimming pool.

• The cup contains a small mass of water (perhaps about 250 g), leading to a proportionate amount of thermal energy.
• In contrast, a swimming pool's mass extends into thousands of kilograms. This significant mass contributes enormously to its overall energy content, overshadowing even a lower temperature.

The formula \( Q = mc\Delta T \), with \( m \) representing mass, helps clarify how mass directly influences the total energy. Even at a lower temperature, the vast water quantity in a pool means it maintains a greater energy sum compared to a small cup. Therefore, mass, in this case, tips the balance heavily towards the pool having more total energy.