Problem 121 - University Physics with Modern Physics

Question

The icecaps of Greenland and Antarctica contain about 1.75\(\%\) of the total water (by mass) on the earth's surface; the occens contain about \(97.5 \%,\) and the other 0.75\(\%\) is mainly groundwater. Suppose the icecaps, currently at an average temperature of about \(-30^{\circ} \mathrm{C},\) somehow slid into the ocean and melted. What would be the resulting temperature decrease of the ocean? Assume that the average temperature of ocean water is currently \(5.00^{\circ} \mathrm{C}\) .

Step-by-Step Solution

Answer

The melting icecaps would decrease ocean temperature.

Step 1: Understand the problem

We need to determine the temperature decrease of the ocean if the icecaps, which are initially at -30°C, melt into the ocean. The problem provides the percentage of water contained in the icecaps and the oceans on Earth's surface.

Step 2: Calculate the relative masses

The icecaps account for 1.75% of the total water on Earth's surface, while the ocean accounts for 97.5%. We'll assume a total mass of water such that these percentages hold. Let the total mass be 100 units for simplicity, yielding an icecap mass of 1.75 units and ocean mass of 97.5 units.

Step 3: Calculate energy for heating ice to 0°C

First, we calculate the energy needed to heat the ice from \(-30^{\circ} C\) to \(0^{\circ} C\). The specific heat capacity of ice is approximately \(c_{ice} = 2.09 \, J/g^\circ C\). Using the formula \(Q = m \, c \, \Delta T\), where \(\Delta T = 30\ \)°C, we find the energy needed.

Step 4: Calculate the energy for phase change

Next, calculate the energy required to melt the ice at 0°C into water at 0°C. The latent heat of fusion for water is about \(L_f = 334 \, J/g\). Use \(Q = m \, L_f\) to find this energy.

Step 5: Calculate total energy added to the ocean

Sum the energy required from Steps 3 and 4. This total energy will be added to the ocean as the ice warms and melts.

Step 6: Calculate temperature change of ocean

The total energy from Step 5 will cause a temperature change in the ocean. Using the specific heat capacity of water \(c_{water} = 4.18\ \,J/g^\circ C\), apply \(Q = mc\Delta T\) to find the temperature decrease \(\Delta T\). Solve for \(\Delta T\) to find the new ocean temperature.

Key Concepts

Phase Change

A phase change occurs when a substance transitions between solid, liquid, or gas states. This process does not necessarily involve a change in temperature but rather a change in the energy state of the substance.
For example, when ice melts to form liquid water, it undergoes a phase change from solid to liquid. During this transformation, energy is absorbed to break the molecular bonds holding the ice structure together.

• Phase changes occur at specific temperatures and pressures.
• The common phase changes include melting, freezing, vaporization, condensation, sublimation, and deposition.

Understanding the concept of phase change is crucial to explaining how ice, when thrown into the ocean, can affect the temperature and composition of ocean water.

Heat Capacity

Heat capacity is a property of a material that indicates how much heat it requires to change its temperature by a certain amount. The higher the heat capacity, the more energy is needed to raise a material's temperature.
Specific heat capacity, often simply referred to as specific heat, is a more relatable concept. It refers to the amount of heat per unit mass required to raise the temperature by one degree Celsius (or Kelvin).

• For instance, the specific heat capacity of ice is about 2.09 J/g°C, meaning it takes 2.09 joules to raise 1 gram of ice by 1°C.
• Water has a significantly higher specific heat capacity, around 4.18 J/g°C, which helps moderate the Earth's climate by absorbing substantial amounts of heat.

Understanding heat capacity helps in calculating the energy exchanges during heating or cooling processes, such as when ice melts in the ocean.

Energy Transfer

Energy transfer is the movement of energy from one place or object to another and can occur in several ways, including conduction, convection, and radiation. In the context of melting ice caps, energy transfer is primarily concerned with heating the ice to transform it into water.
When energy is transferred to ice, it causes the ice to warm until it reaches the melting point. At this point, any additional energy contributes to the phase change from solid to liquid, rather than increasing temperature.

• Exchange of energy during phase changes does not increase temperature; instead, it changes the state of the matter.
• This process is important for understanding how the ice caps melting into the ocean would lead to a decrease in the overall temperature of the ocean.

Recognizing the idioms of energy transfer aids in comprehending the thermal dynamics at play during such large-scale environmental processes.

Latent Heat

Latent heat refers to the heat required to change the state of a substance without altering its temperature. In our example of ice melting, latent heat is the energy needed to convert solid ice at 0°C to liquid water at the same temperature.
The latent heat of fusion is specifically the energy needed for a solid to become a liquid. For water, this is approximately 334 J/g.

• Latent heat varies between substances based on molecular composition and bonding.
• It is crucial for understanding processes like ice melting into water, as the latent heat must be supplied to affect the phase change.

Grasping the concept of latent heat is essential for solving problems related to phase changes and energy balance, like predicting temperature shifts caused by melting ice caps.