Problem 168 - General Chemistry

Question

Antarctica, almost completely covered in ice, has an area of \(5,500,000 \mathrm{mi}^{2}\) with an average height of \(7500 \mathrm{ft}\). Without the ice, the height would be only \(1500 \mathrm{ft}\). Estimate the mass of this ice (two significant figures). The density of ice is \(0.917 \mathrm{~g} / \mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

The mass of the ice is approximately 2.4 × 10^15 kg.

Step 1: Identify extra ice thickness

First, we need to calculate the thickness of the ice covering Antarctica. Given that the average height with ice is 7500 ft and without ice, it would be 1500 ft, the extra height due to ice is: 7500 ft - 1500 ft = 6000 ft.

Step 2: Convert thickness to compatible units

Next, convert the ice thickness from feet to centimeters, as the density of ice is given in grams per cubic centimeter. There are 30.48 centimeters in a foot. Therefore, 6000 ft equals 6000 ft × 30.48 cm/ft = 182880 cm.

Step 3: Convert area to compatible units

Now, convert the area of Antarctica from square miles to square centimeters. There are 2.59 × 10^6 square centimeters in a square mile. So, the area is 5,500,000 mi² × 2.59 × 10^6 cm²/mi² = 1.4245 × 10^13 cm².

Step 4: Calculate volume of ice

The volume of the ice is found by multiplying the area by the thickness, both in compatible units. Thus, the volume of ice is 1.4245 × 10^13 cm² × 182880 cm = 2.603772 × 10^18 cm³.

Step 5: Calculate mass of ice

Finally, calculate the mass of the ice using the formula: mass = volume × density. The density of ice is 0.917 g/cm³, so the mass is 2.603772 × 10^18 cm³ × 0.917 g/cm³ = 2.388066364 × 10^18 grams. Converting to kilograms (since there are 1000 grams in a kilogram), the mass is 2.388066364 × 10^15 kg, which rounds to 2.4 × 10^15 kg for two significant figures.

Key Concepts

Density of Ice

Density is a measure of how much mass is contained in a given volume. For ice, this is typically valued at 0.917 grams per cubic centimeter (g/cm³). This specific density of ice is important in scientific calculations, such as estimating the mass of large ice quantities, like Antarctica.

Remember that density can vary slightly due to temperature fluctuations, but for this exercise, we stick to the standard density which proves handy to simplify calculations. Density, symbolized as \( \rho \), is calculated with the formula:
\[ \rho = \frac{m}{V} \]where \( m \) is mass and \( V \) is volume. Thus, understanding density allows us to derive mass from volume directly. Just multiply the volume of ice by the density to find the total mass.

It’s essential to take care when handling units, as density could be provided in different measures, depending on the context.

Volume of Ice

Volume is the amount of space that a substance occupies. In our exercise, correctly calculating the volume of ice covering Antarctica is crucial for estimating its total mass.

In the problem, the area of Antarctica is given as 5,500,000 square miles. It's as if we are spreading this area over a certain thickness, which is calculated as the difference in height with and without ice, amounting to 6000 feet.

• Calculate the volume, by multiplying the area by the thickness, ensuring both are in the same unit. This gives
  \[ \text{Volume} = \text{Area} \times \text{Thickness} \]
• It's vital to perform unit conversions to ensure the calculation is accurate and practical.

To get the volume in cubic centimeters, first convert all measures to centimeters since the density is provided in grams per cubic centimeter.

Unit Conversion

Unit conversion is the process of changing the units of a measurement to another. It ensures that calculations like these can be simplified and precise, allowing formulas to be used effectively without unit discrepancies.

• Convert the thickness of the ice from feet to centimeters, knowing that 1 foot equals 30.48 centimeters. Thus:
  \[6000 \text{ ft} \times 30.48 \text{ cm/ft} = 182880 \text{ cm}\]
• For the area, convert square miles to square centimeters, using the conversion factor that 1 square mile equals \( 2.59 \times 10^6 \) square centimeters.
  \[5,500,000 \text{ mi}^2 \times 2.59 \times 10^6 \text{ cm}^2/\text{mi}^2 = 1.4245 \times 10^{13} \text{ cm}^2 \]

These conversions are necessary to keep all measurements compatible, as we then proceed to calculate the volume and mass.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. When performing calculations, it’s crucial to use consistent significant figures to prevent errors in measurement precision. In the calculation of the mass of Antarctic ice:

• The numerical result initially found was refined to a number maintaining two significant figures, required by the problem statement.
• When dealing with large numbers, rounding can introduce potential inaccuracies, making the understanding of significant figures even more necessary.
• In cases such as this, rounding must always be done with caution - checking that outcomes preserve their scientific relevance and practicality.

Ensuring correct use of significant figures is indispensable in scientific and mathematical calculations to communicate data precisely.

Geographic Measurements

Geographic measurements often deal with large areas and changes in elevation and require conversions to connect to other scientific data.

In this problem, having the initial area and height of Antarctica, both with and without ice cover, provides a real-world application of geographical measurement uses.

• The large-scale area (5,500,000 square miles) underlies the importance of using appropriate significant figures.
• Changes in geographic features like height (7500 ft with ice vs. 1500 ft without) are crucial for proper calculations in scientific geography.

Such geographic measurements when combined with density, give a complete picture of phenomena like ice coverage and climate data analysis. They enable a bridge between field observations and theoretical estimations, akin to the ice mass calculation problem.