Problem 75 - Chemistry: Atoms First

Question

A long ton is defined as exactly 2240 lb. What is this mass in kilograms?

Step-by-Step Solution

Answer

A long ton (2240 lb) is equal to approximately 1016 kilograms.

Step 1: Understand the conversion relationship

First, we need to recall the conversion factor between pounds (lb) and kilograms (kg). The conversion factor is that 1 pound is equivalent to 0.45359237 kilograms.

Step 2: Convert pounds to kilograms

To convert from pounds to kilograms, multiply the number of pounds by the conversion factor (0.45359237 kg/lb). So we have to calculate 2240 lb multiplied by 0.45359237 kg/lb.

Step 3: Perform the multiplication

Multiplying 2240 lb by 0.45359237 kg/lb gives us the mass in kilograms. The calculation is as follows: \( 2240 \text{ lb} \times 0.45359237 \frac{\text{kg}}{\text{lb}} = 1016.0469088 \text{ kg} \).

Step 4: Round the result (if necessary)

In practice, we would typically round the result to a reasonable number of significant figures. Here, we could round the mass in kilograms to three significant figures: \( 1016 \text{ kg} \).

Key Concepts

Conversion Factor

Understanding the concept of a conversion factor is the backbone of many scientific computations, particularly in chemistry. A conversion factor is a ratio that expresses how many of one unit are equal to another unit. For example, we know that 1 pound is equal to 0.45359237 kilograms.

This number isn't arbitrary; it is an exact calculated value that allows us to accurately change measurements from pounds to kilograms without changing the quantity of what we're measuring. Conversion factors are built from equivalences and are used in equations to cancel out units until we are left with the desired outcome. In chemical calculations, these factors are essential for dealing with units like moles, liters, and grams and enable us to work within the metric system, which is the standard in most scientific work.

Pounds to Kilograms

Let's dive deeper into the practical example of converting pounds to kilograms. The problem gives us a mass in pounds—the long ton, which is precisely 2240 lb. To convert this to kilograms, we utilize our previously mentioned conversion factor. Multiply 2240 by 0.45359237, as one pound is equivalent to 0.45359237 kilograms.

This conversion is necessary in many real-world situations, especially those involving international trade, science, and technology, where the metric system is predominantly used. By having a standard conversion factor, we can ensure clear communication and prevent misunderstandings due to measurement differences.

Significant Figures

After performing the conversion, we obtain a result that can have many decimal places. In scientific practice, these decimals often need to be rounded to reflect the precision in our measurements - this is where significant figures come into play.

Significant figures are the digits in a number that carry meaningful contributions to its precision. This includes all non-zero digits, zeros between non-zero digits, and any final zeros to the right of the decimal point. In our example, the exact result was 1016.0469088 kg, but when we round to three significant figures, we get 1016 kg. Rounding to the correct number of significant figures is crucial to ensure that we don't overstate the precision of our calculated values.

Why Three Significant Figures?

Typically, the number of significant figures we round to correlates with the precision of the input data. In this case, '2240 lb' has four significant figures (assuming the zeros are measured values), but if we were to express our final answer (in common practice), rounding to three significant figures is sufficient to convey the necessary precision without adding excessive detail.