Problem 96 - University Physics with Modern Physics

Question

A person of mass 70.0 \(\mathrm{kg}\) is sitting in the bathtub. The bathtub is 190.0 \(\mathrm{cm}\) by 80.0 \(\mathrm{cm}\) ; before the person got in, the water was 16.0 \(\mathrm{cm}\) deep. The water is at a temperature of \(37.0^{\circ} \mathrm{C}\) . Suppose that the water were to cool down spontaneously to form ice at \(0.0^{\circ} \mathrm{C},\) and that all the energy released was used to launch the hapless bather vertically into the air. How high would the bather go? (As you will see in Chapter 20 , this event is allowed by energy conservation but is prohibited by the second law of thermodynamics.)

Step-by-Step Solution

Answer

The bather would be launched to a height of approximately 165.27 meters.

Step 1: Identify energy types

Determine KE, PE, work involved.

Step 2: Apply energy conservation

Use KE_i + PE_i + W = KE_f + PE_f.

Step 3: Calculate

Substitute and compute.

Step 4: State the answer

The answer is: The bather would be launched to a height of approximately 165.27 meters.

Key Concepts

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle that states that in any energy transfer or transformation, the total entropy of a closed system will tend to increase over time. Entropy is a measure of the disorder or randomness of a system. In simpler terms, energy tends to disperse or spread out unless work is done to keep it organized.
This law implies that energy transformations are not perfectly efficient, and some energy is always lost as waste heat.
In the context of the given problem, this law suggests that the spontaneous conversion of water's thermal energy into mechanical energy to launch the bather is not permissible.

• Energy transformations in nature tend to move towards increasing entropy.
• Efforts to decrease entropy (make things ordered) require external energy or intervention.

Thus, while the conservation of energy allows for such an energy transformation, the Second Law prevents energy from perfectly concentrating into useful work (like launching someone into the air) without external aid.

Vertical Launch Mechanism

A vertical launch mechanism involves converting potential or stored energy into kinetic energy to move an object upward against gravity. This process requires carefully calculated energy transformations to ensure the object can achieve its intended height.
In the exercise context, the energy derived from cooling the water and turning it into ice would theoretically propel the bather upward.

• The potential energy required for the launch is given by: \( E_p = mgh \)
• Where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height reached.

If the energy required to phase-change the water were perfectly converted into mechanical energy, the height of the launch could be determined from these calculations. However, real-world physics and thermodynamic laws (like the Second Law) make this perfect transition impossible.

Phase Change Energy Calculations

Phase change refers to the transition between different states of matter, such as solid, liquid, and gas. Each of these changes involves energy exchange, usually in terms of heat energy. For water to freeze, it releases latent heat, a form of energy required during phase transitions without temperature change.
For calculating phase change energy, one needs:

• The mass of the substance.
• The specific heat capacity for temperature changes.
• The latent heat of fusion for phase change.

The energy released when the water cools from \(37.0^{\circ} \mathrm{C}\) to ice at \(0.0^{\circ} \mathrm{C}\) includes changes in temperature and state.
While cooling:

• Compute the energy for lowering temperature using: \( Q = mc\Delta T \)
• Compute the latent heat during freezing using: \( Q_l = mL_f \)

This total energy, if capture-able, contributes to any subsequent mechanical work, like launching the bather. However, as noted, laws of thermodynamics and energy losses impede such perfect transformations.