Problem 167 - General Chemistry

Question

The land area of Greenland is \(840,000 \mathrm{mi}^{2},\) with only \(132,000 \mathrm{mi}^{2}\) free of perpetual ice. The average thickness of this ice is \(5000 \mathrm{ft}\). Estimate the mass of the ice (assume 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 \(2.6 \times 10^{12}\) metric tons.

Step 1: Calculate Ice-Covered Area

First, determine the area covered by ice. Subtract the ice-free area from the total land area of Greenland: \( 840,000 \, \text{mi}^2 - 132,000 \, \text{mi}^2 = 708,000 \, \text{mi}^2 \). This is the area covered by ice.

Step 2: Convert Area to Square Feet

Convert the ice-covered area from square miles to square feet. There are \(5280\) feet in a mile, so:\[708,000 \, \text{mi}^2 \times (5280 \, \text{ft/mi})^2 = 19,730,688,000,000 \, \text{ft}^2.\]

Step 3: Calculate Volume of Ice

Multiply the ice-covered area in square feet by the average thickness of the ice to find the volume in cubic feet:\[19,730,688,000,000 \, \text{ft}^2 \times 5000 \, \text{ft} = 98,653,440,000,000,000 \, \text{ft}^3.\]

Step 4: Convert Volume to Cubic Centimeters

Convert the volume from cubic feet to cubic centimeters. There are \(30.48\) centimeters in a foot, so:\[98,653,440,000,000,000 \, \text{ft}^3 \times (30.48 \, \text{cm/ft})^3 = 2.793632 \, \times 10^{18} \, \text{cm}^3.\]

Step 5: Calculate Mass of Ice

Use the density of ice to calculate the mass in grams:\[2.793632 \times 10^{18} \, \text{cm}^3 \times 0.917 \, \text{g/cm}^3 = 2.561018 \times 10^{18} \, \text{g}.\]

Step 6: Convert Mass to Appropriate Units

Convert the mass from grams to metric tons for a more understandable figure (1 metric ton = 10^6 grams):\[2.561018 \times 10^{18} \, \text{g} \div 10^6 = 2.561018 \times 10^{12} \, \text{metric tons}.\]

Step 7: Apply Significant Figures

Round the final answer to two significant figures since the data provided mostly follows two significant figures:\[2.6 \times 10^{12} \, \text{metric tons}.\]

Key Concepts

Density Calculation

Density is a key property in chemistry that defines how much mass is contained in a given volume. It is often represented by the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This formula specifies that if you know the volume of an object and its density, you can easily calculate its mass.

• The exercise you are looking at provides the density of ice: \(0.917 \text{g/cm}^3\).
• Once you have the volume of the ice, multiplying it by this density gives you the mass of the ice.

The calculated volume in the solution is in cubic centimeters, which is compatible with the density unit \(\text{g/cm}^3\). This compatibility is crucial, as it ensures that the units cancel out properly, giving you the mass in grams. The understanding of density's importance is foundational in solving many chemistry and physics-related problems.

Unit Conversion

Unit conversion is a method used to express physical quantities in different units of measurement. In this exercise, there are several conversions performed.First, converting the ice-covered area from square miles to square feet involves understanding that

• 1 mile = 5280 feet
• This means 1 square mile is equal to \( (5280)^2 \) square feet.

You multiply the area of \(708,000 \text{mi}^2\) by \((5280 \text{ ft/mi})^2\) to get the area in square feet.Next, the conversion of volume from cubic feet to cubic centimeters is done by using the conversion factor: 1 foot = 30.48 cm. Therefore,

• 1 cubic foot is \((30.48)^3\) cubic centimeters.

Executing these conversions correctly ensures that all subsequent calculations have consistent and correct units.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a concise form, making them easier to read and use in calculations. In scientific notation, numbers are expressed as a product of a number between 1 and 10 and a power of 10.During the volume conversion, the result is an astronomically large number: \(98,653,440,000,000,000 \text{ft}^3\), which is expressed as

• \(2.793632 \times 10^{18} \text{cm}^3\) in scientific notation.

This format makes it easier to handle the number in further mathematical operations such as multiplication and division.Scientific notation is particularly useful in chemistry as it helps in managing numbers commonly encountered when dealing with the massive or minuscule scales of quantities in the physical sciences.

Significant Figures

Significant figures are the digits in a measurement or calculation that contribute to its precision. The rules for significant figures are critical for determining the degree of uncertainty in the final result.In this exercise, the data predominantly maintains two significant figures:

• The density of ice: \(0.917 \text{g/cm}^3\)
• The land areas given: \(840,000 \text{mi}^2\) and \(132,000 \text{mi}^2\).

Thus, when rounding the calculated mass of the ice to fit this level of precision, the answer becomes:

• \(2.6 \times 10^{12} \text{metric tons}\)

Being aware of the significant figures ensures that the numerical confidence in the data is upheld throughout the computation process.