Question

The total volume of ice in the Antarctic is about \(3.01 \times 10^{7} \mathrm{km}^{3} .\) If all the ice in the Antarctic were to melt completely, estimate the rise, \(h,\) in sea level that would result from the additional liquid water entering the oceans. The densities of ice and fresh water are \(0.92 \mathrm{g} / \mathrm{cm}^{3}\) and \(1.0 \mathrm{g} / \mathrm{cm}^{3},\) respectively. Assume that the oceans of the world cover an area, \(A,\) of about \(3.62 \times 10^{8} \mathrm{km}^{2}\) and that the increase in volume of the oceans can be calculated as \(A \times h\).

Short answer

The sea level would rise approximately 67.32 m if all Antarctic ice were to melt completely.

Step-by-step solution

Calculate the Total Volume of Melted Ice

Since 1 g of ice melts to 1 g of water, and the densities of ice and water are 0.92 g/cm³ and 1.0 g/cm³ respectively, we can find the volume of water produced when all the ice melts by using the formula: \[ V_{water} = V_{ice} \times \frac{Density_{ice}}{Density_{water}} \] where \(V_{water}\) is the volume of water, \(V_{ice}\) is the volume of ice, \(Density_{ice}\) is density of ice and \(Density_{water}\) is the density of water. Heap in mind that initial volume of ice is given in km³ and we need to convert it to cm³ for consistency with density units. 1 km³ = \(10^{15}\) cm³.

Calculate the Rise in Sea Level

The volume of the resulting water will spread around the world's oceans and the sea level will rise accordingly. This rise in sea level, \(h\), can be calculated by dividing the volume of water, \(V_{water}\), by the area of the oceans, \(A_ocean\). Hence: \[h = \frac{V_{water}}{A_{ocean}}\] Again, bear in mind the units. The area of the oceans is given in km² , thus we should perform a conversion from km² to cm²: 1 km² = \(10^{10}\) cm² .

Substituting values

Now we substitute the given values into formulas outlined in steps 1 and 2 and calculate the final answer. Remembering about all necessary unit conversions.

Check the Answer

Finally, as common practice, it's always important to make sure the answer makes sense in the context of the problem. In this case, double-checking whether calculated value for sea level rise does seem plausible given the known data about the Earth's hydrosphere.

Key concepts

Density of Ice

Density is a way to measure how much mass is contained in a unit volume of a material. For ice, this means understanding how tightly its molecules are packed together. Scientists often use the unit grams per cubic centimeter (g/cm³) to express density. Ice has a density of 0.92 g/cm³.

Because ice is less dense than water, it floats. This is why icebergs have only a small part visible above the water's surface. When you know the density of ice, you can predict how much water it would convert into when melted, simply because density is a comparative measure. For instance, the density of ice (0.92 g/cm³) allows us to deduce that it has 92% of the density of water. This is crucial in determining the volume of water that results when ice melts.

Density of Water

Water is denser than ice, with a density of 1.0 g/cm³. This higher density means that water molecules are packed closer together compared to ice molecules. The reason why water is denser than ice is because of the structure of water molecules that are tightly bonded in liquid form.

The density of water helps in calculating how ice turns into water. With the density information, one can determine changes in volume as the state changes from solid (ice) to liquid (water). By understanding these densities, we can solve questions related to the melting ice and how much liquid water would be produced.

Volume Conversion

Volume conversion is an essential part of solving problems involving changes in states of matter, such as solid to liquid. In our context, we are dealing with very large volumes initially given in cubic kilometers (km³).

To link the densities expressed in grams per cubic centimeter to these large volume units, we need to perform conversions. Specifically, we convert volumes of ice from km³ to cm³. There are a staggering number of cubic centimeters in a cubic kilometer: 1 km³ = \(10^{15}\) cm³. Carefully handling unit conversion helps maintain consistency in calculations.

Antarctic Ice Volume

The Antarctic contains a massive amount of ice, about \(3.01 \times 10^{7}\) km³. This volume represents a critical component of the Earth's long-term climate balance. When ice melts, it contributes to sea level rise, a significant concern for global climates.

To estimate the potential sea level rise if all this ice melted, one must calculate the additional water volume added to the oceans and understand how this additional volume would spread over the global ocean area. By applying the principles of density and volume, as well as conversion formulas, we can accurately predict sea level changes, connecting local ice volumes to global impacts.