Problem 84 - College Physics

Question

\(\bullet\) The icecaps of Greenland and Antarctica contain about 1.75\(\%\) of the total water (by mass) on the earth's surface; the oceans 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 ocean's temperature would decrease by approximately \(\Delta T\) degrees Celsius.

Step 1: Identify Known Quantities

Let's identify and list the known facts from the problem. The icecaps account for 1.75% of the total water, the oceans account for 97.5%, and the average current ocean temperature is 5.00°C. The icecaps are at a temperature of -30°C when they melt.

Step 2: Calculate Mass Proportions of Water Types

Since the problem doesn't specify the total mass of water, assume a placeholder mass \(M\). The mass of the ocean is \(0.975M\) and the mass of the icecaps is \(0.0175M\).

Step 3: Find Heat Transfer Required

Calculate the heat required to change the temperature of the icecaps from -30°C to 0°C (melting point), and then to raise the melted water to 5°C. Use the formula for heat transfer: \(Q = mc\Delta T\).

Step 4: Heat Needed for Icecaps to Melt and Warm Up

The specific heat of ice is \(c_{ice} = 2.09 \text{ J/g°C}\), and for water \(c_{water} = 4.18 \text{ J/g°C}\). Heat needed to raise temperature from -30°C to 0°C is \[Q_1 = 0.0175M \times 1000 \text{ g/kg} \times 2.09 \times (0-(-30))\]. After melting, heat to warm the water to 5°C: \[Q_2 = 0.0175M \times 1000 \text{ g/kg} \times 4.18 \times (5-0)\].

Step 5: Calculate Total Heat Absorbed by Ocean

The latent heat of fusion of ice is 334 J/g. Calculate the total heat absorbed: \[Q_{total} = Q_1 + 0.0175M \times 1000 \text{ g/kg} \times 334 + Q_2\].

Step 6: Determine Change in Temperature of Ocean

The total heat lost by the ocean, \(Q_{lost} = Q_{total}\). Use the formula \(Q = mc\Delta T\) for the ocean water: \[\Delta T = \frac{Q_{total}}{0.975M \times 1000 \text{ g/kg} \times 4.18}\]. Calculate \(\Delta T\).

Step 7: Conclusion

The change in temperature calculated in Step 6 gives the decrease in ocean temperature due to melting icecaps. Utilizing the conservation of energy, the additional water masses will decrease the ocean's initial temperature. Solve for this change.

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics. It refers to the movement of thermal energy from one object or material to another, driven by a temperature difference. In our exercise, we explore how the heat transfer from the ocean to the melting icecaps affects the temperature of the ocean.

There are three common methods of heat transfer: conduction, convection, and radiation. However, in this context, it's the natural heat exchange between the ice and water that plays the key role. Heat moves from the relatively warmer ocean to the much colder icecaps until a thermal equilibrium is reached.

• When the icecaps drop into the ocean, they absorb heat from the water as they melt.
• This process causes the overall ocean temperature to drop since thermal energy from the water is used to melt the ice.
• Understanding this exchange is crucial to predicting the environmental impact of large-scale melting icecaps.

Specific Heat Capacity

Specific heat capacity is an important property of materials and substances, indicating the amount of heat needed to change their temperature. It tells us how a particular substance will respond to adding or removing heat.

The formula for calculating heat using specific heat capacity is: \[ Q = mc\Delta T \] where:

• \(Q\) is the heat absorbed or released,
• \(m\) is the mass of the substance,
• \(c\) is the specific heat capacity, and
• \(\Delta T\) is the change in temperature.

In our exercise, different specific heat capacities are considered:

• Icy water has a specific heat of \(2.09 \, \text{J/g°C}\), reflecting its resistance to temperature change.
• Liquid water, post-melting, has \(4.18 \, \text{J/g°C}\), showing it can hold more heat.

This means a larger amount of energy is required to change the temperature of water compared to ice, illustrating how much heat is absorbed before the ocean's overall temperature significantly drops.

Melting Ice Caps

The melting ice caps scenario describes what happens when large masses of ice slide into the ocean and reach their melting point. This process significantly impacts the ocean's thermal balance.

Globally, this has profound implications:

• The addition of melted water alters ocean circulation and weather patterns.
• It raises sea levels, potentially affecting coastal regions around the world.
• The temperature equilibrium of the ocean shifts, affecting marine ecosystems.

In our exercise, ice melts from \(-30^{\circ} \mathrm{C}\) to \(0^{\circ} \mathrm{C}\), consuming energy for both reaching the melting point and turning into water. Surfaces like ice caps have high albedo, meaning they reflect more sunlight, thus interacting differently with energy compared to when they melt. An understanding of these dynamics is crucial in the study of climate change, emphasizing the importance of predicting and mitigating these vast environmental changes.

Latent Heat of Fusion

The latent heat of fusion refers to the energy required to change a substance from a solid to a liquid state at its melting point without altering its temperature. This concept is crucial in thermodynamics for understanding state changes.

The latent heat of fusion for ice is \(334 \, \text{J/g}\), a measure of how much heat energy is utilized in the melting process without raising the temperature of the ice itself. During this transformation:

• The absorbed energy breaks the molecular bonds in ice, turning it into liquid water.
• Despite the absorption, no temperature change is noted until all ice becomes water.

In our exercise, calculating the latent heat contributes to the total heat absorbed from the ocean. It helps quantify the impact on the ocean's temperature as additional energy is required beyond merely raising temperatures. Comprehending latent heat is essential for appreciating the vast energetic exchanges involved in environmental processes like melting ice caps.