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Global warming. As the earth warms, sea level will rise due to melting of the polar ice and thermal expansion of the oceans. Estimates of the expected temperature increase vary, but \(3.5 \mathrm{C}^{\circ}\) by the end of the century has been plausibly suggested. If we assume that the temperature of the oceans also increases by this amount, how much will sea level rise by the year 2100 due only to the thermal expansion of the water? Assume, reasonably, that the ocean basins do not expand appreciably. The average depth of the ocean is \(4000 \mathrm{~m}\), and the coefficient of volume expansion of water at \(20^{\circ} \mathrm{C}\) is \(0.207 \times 10^{-3}\left(\mathrm{C}^{\circ}\right)^{-1}\)

Step by step solution

01

Understand the Variables

To solve this exercise, we need to determine how much the sea level will rise due to the thermal expansion of the water. We are given the average depth of the ocean as \( 4000\, \mathrm{m} \), the temperature increase \( \Delta T = 3.5\, \mathrm{C}^{\circ} \), and the coefficient of volume expansion of water as \( \beta = 0.207 \times 10^{-3} (\mathrm{C}^{\circ})^{-1} \).

02

Apply the Formula for Thermal Expansion

The change in volume \( \Delta V \) due to thermal expansion is given by the formula:\[ \Delta V = \beta \times V_0 \times \Delta T \]where \( V_0 \) is the initial volume. Since we want the change in height (sea level rise \( \Delta h \)), we can express \( \Delta V \) as:\[ \Delta V = A \times \Delta h \]where \( A \) is the surface area of the ocean.

03

Express Change in Volume in Terms of Change in Height

Assuming the surface area of the ocean remains constant, the volume expansion can be equated to the product of surface area and the change in height (sea level rise):\[ V_0 \times \beta \times \Delta T = A \times \Delta h \] Since \( V_0 = A \times 4000 \), we can rearrange this to find \( \Delta h \): \[ A \times 4000 \times \beta \times \Delta T = A \times \Delta h \] This simplifies to:\[ \Delta h = 4000 \times \beta \times \Delta T \].

04

Calculate the Change in Sea Level

Substitute the given values into the equation:\[ \Delta h = 4000 \times 0.207 \times 10^{-3} \times 3.5 \]\[ \Delta h = 4000 \times 0.207 \times 3.5 \times 10^{-3} \]\[ \Delta h = 2.898 \mathrm{m} \].

05

Interpret the Result

The calculated change in sea level \( \Delta h = 2.898 \mathrm{m} \) indicates that by the year 2100, the sea level is expected to rise due to thermal expansion by approximately 2.90 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Global Warming and Its Impact

Global warming refers to the gradual increase in Earth's average surface temperature, primarily due to greenhouse gas emissions like carbon dioxide and methane. Human activities, such as burning fossil fuels and deforestation, accelerate this process.
Major consequences of global warming include:

• Rising temperatures
• Increased occurrence of extreme weather events
• Changes in precipitation patterns
• Sea level rise

All of these changes significantly impact various ecosystems, human health, and economies globally. With global temperatures forecasted to rise by 3.5°C by the century's end, understanding its repercussions becomes crucial for future planning.

Sea Level Rise Explained

The rise in sea levels is largely driven by two main factors: melting ice caps and glaciers, and the thermal expansion of seawater as it warms. When water heats up, it takes up more space, contributing to sea level rise.

• A warmer climate melts ice, adding more freshwater to the ocean.
• Thermal expansion causes existing ocean water to occupy more volume.

Given that the oceans cover 71% of Earth’s surface and hold vast volumes of water, even small increases in volume can lead to significant changes in sea level. Coastal areas around the world are at risk, possibly facing increased flooding and erosion due to these rising sea levels.

Understanding the Coefficient of Volume Expansion

The coefficient of volume expansion is a crucial parameter in understanding how materials respond to temperature changes. It quantifies the fractional change in volume per degree Celsius of temperature increase.
For water, this coefficient is approximately 0.207 x 10^-3 (°°ä)^-1 at 20°C. This means that for each degree Celsius rise in temperature, the volume of water expands by a small fraction relative to its initial volume.
Applying this concept to the oceans helps us predict how much the sea level will rise due to thermal expansion with a given increase in temperature. Precise knowledge of this coefficient allows scientists to model future sea-level changes with higher accuracy.

Temperature Increase Effect on Oceans

When the temperature of the oceans increases due to global warming, several effects come into play:

• Thermal Expansion: As previously mentioned, warmer water expands, raising sea levels.
• Changes in Ocean Currents: Temperature changes can alter ocean currents, impacting global climate and marine life.
• Coral Bleaching: Elevated temperatures can lead to coral bleaching, where corals lose their vibrant colors and vital algae.

The expected 3.5°C temperature rise by 2100 will significantly impact these factors, highlighting the urgent need for climate action to mitigate potential damage to ocean ecosystems and coastal communities.