Problem 44 - Chemistry

Question

Which of the following statements use exact numbers? (a) \(1 \mathrm{ft}=12 \mathrm{in}\). (b) \(1 \mathrm{cal}=4.184 \mathrm{~J}\) (c) The height of Mt. Everest is \(29,035 \mathrm{ft}\). (d) The world record for the 1 mile run, set by Morocco's \(\mathrm{Hi}\) cham el Guerrouj in July, 1999 , is 3 minutes, \(43.13\) seconds.

Step-by-Step Solution

Answer

Statement (a) uses exact numbers.

Step 1: Define Exact Numbers

Exact numbers are those that are known with complete certainty without any approximation. They are often related to definitions or counting.

Step 2: Analyze Statement (a)

Statement (a), \(1 \mathrm{ft}=12 \mathrm{in}\), is a defined relationship based on the definition of a foot being equal to 12 inches. This is exact because it is a standard definition.

Step 3: Analyze Statement (b)

Statement (b), \(1 \mathrm{cal}=4.184 \mathrm{~J}\), involves a conversion factor derived through measurement. This is not considered exact due to the decimal and precision involved.

Step 4: Analyze Statement (c)

Statement (c), The height of Mt. Everest is \(29,035 \mathrm{ft}\), is based on a measurement, and measurements are never exact due to possible measurement errors and limitations in precision.

Step 5: Analyze Statement (d)

Statement (d), The world record is 3 minutes, \(43.13\) seconds, is also based on a measurement with precision (decimal), which makes it inexact.

Key Concepts

Measurement and Precision

In the world of chemistry and science, measurements are vital but inherently come with limitations. Measurements are never exact because there is always some degree of uncertainty and potential for error involved. When we measure anything, whether it's the height of a mountain or the time it takes to complete a race, we rely on instruments that have certain limitations.

These instruments are only accurate to a certain level, beyond which the results become less reliable. This imprecision arises from several factors:

• Instrument precision and calibration errors.
• Human interpretation and recording errors.
• External conditions affecting the measurement process.

Understanding this helps us appreciate why, for example, the height of Mt. Everest cannot be stated with absolute certainty. Precision of measurement highlights the inherent variability and helps in planning more accurate experiments and calculations.

Defined Relationships

Defined relationships are essential in science as they provide a foundation of exactness. These are numbers and relationships that have been established universally and accepted without ambiguity. For example, knowing that 1 foot is exactly 12 inches is a defined relationship. It was set by definition and carries no uncertainty because it is universally accepted and agreed upon.

Such defined relationships do not change regardless of the context and are free from the vagaries of measurement. They typically arise in cases:

• Where units are defined in relation to each other, like feet and inches.
• Where constants are used in mathematical formulas.
• In counting numbers, such as the number of eggs in a dozen.

These relationships form the backbone of scientific calculations, allowing scientists to perform precise and repeatable processes without the combined uncertainty of measurement.

Conversion Factors in Chemistry

Conversion factors are tools that allow us to switch between different units of measurement. While they facilitate important calculations, not all conversion factors are exact. Some, like the energy conversion example of 1 calorie equaling 4.184 Joules, are derived from experimental results and are subject to the limits of precision and accuracy inherent in those measurements.

In chemistry, conversion factors are crucial for balancing equations and converting between units such as moles, liters, and grams. However, it's important to recognize when these factors involve approximations:

• They often carry decimal values indicating precision limits.
• They depend on experimental setups and validation over time.
• Conversion constants may differ slightly based on new methodologies or standards.

Distinguishing exact numbers from measurement-based conversion factors ensures more accurate scientific endeavors and avoids the errors introduced by assuming precision where it does not exist.