Problem 101 - Chemistry A Molecular Approach

Question

The average U.S. farm occupies 435 acres. How many square miles is this? \(\left(1\right.\) acre \(\left.=43,560 \mathrm{ft}^{2}, 1 \mathrm{mile}=5280 \mathrm{ft}\right)\) MISSED THIS? Read Section 1.8; Watch \(\mathrm{KCV} 1.8,\) IWE 1.9

Step-by-Step Solution

Answer

The farm occupies \( 0.68 \quad \text{square miles} \).

Step 1: Convert acres to square feet

To convert the farm size from acres to square feet, multiply the number of acres by the conversion factor (1 acre = 43,560 square feet).

Step 2: Calculate farm size in square feet

Multiply the farm size in acres by the conversion factor to get the farm size in square feet: 435 acres * 43,560 square feet/acre.

Step 3: Convert square feet to square miles

To convert square feet to square miles, divide the farm size in square feet by the square of the conversion factor from miles to feet (1 mile = 5,280 feet), so we use (5,280 feet x 5,280 feet).

Step 4: Calculate farm size in square miles

Divide the farm size in square feet by the conversion factor squared to get the farm size in square miles: (435 acres * 43,560 square feet/acre) / (5,280 feet/mile * 5,280 feet/mile).

Step 5: Simplify the calculation

Perform the calculation to simplify to the final answer in square miles.

Key Concepts

Dimensional Analysis

Dimensional analysis, often known as the factor-label method or the unit factor method, is a cornerstone of problem-solving in fields such as chemistry and physics. It's a way to convert one unit of measurement to another using conversion factors that are known to be equivalent.

Consider the given problem: converting acres to square miles. We approach it by multiplying the original measurement (the number of acres) by a series of conversion factors that cancel out the unwanted units and leave us with the desired units. This method ensures that we maintain consistency within our measurements and allow us to seamlessly navigate complex conversions.

For students, becoming proficient in dimensional analysis requires practice. It's important to keep track of units throughout the process, as careful attention to detail will prevent calculation errors. Moreover, understanding the relationship between different units is central to mastering this concept.

Conversion Factors

Conversion factors are the building blocks of dimensional analysis. They are ratios that express how many of one unit is equivalent to another. For instance, in our agricultural land size problem, the given conversion factors were: 1 acre equals 43,560 square feet, and 1 mile equals 5,280 feet.

By using these conversion factors, you can convert units by multiplying or dividing, depending on whether you want to scale up or down. It is essential to arrange them in such a way that units you wish to discard cancel each other out. For clarity in problem-solving, always write down each step, including units, when applying conversion factors, which can help avoid mistakes and make it easier to follow your work.

Problem-Solving in Chemistry

Problem-solving in chemistry often involves understanding and converting between different units of measurement. A systematic approach is key to simplifying complex problems. Start by identifying the known quantities and the units they're in, and then determine the units you want to end up with.

Next, map out a path using conversion factors to guide each step towards the desired units, keeping in mind to cancel out the units that are no longer needed. In our example problem, we converted acres to square feet and then to square miles through clear steps, each time reducing the complexity of the problem until we reached the solution.

This method not only helps solve textbook problems but also develops critical thinking and analytical skills that are invaluable in academic and professional settings.

Units of Measurement

Units of measurement are standard quantities used to express and measure physical quantities. In the context of the exercise, acres and square miles are units of area. Different systems of units are used around the world, though scientists typically use the International System of Units (SI).

In our example, it is important to be familiar with both the US customary units (acres and square miles) and the conversion process to other units of measurement. Building familiarity with different systems of units allows for versatile problem-solving abilities in chemistry and other scientific disciplines.

Moreover, understanding the relative sizes of different units helps contextualize the measures in practical terms, providing a deeper comprehension beyond the numerical results. For example, conceptualizing the size of an acre in relation to a square mile can aid in visualizing the scale of a given problem or solution.