Problem 129 - Introductory Chemistry Atoms First

Question

Which one of the following expresses the measured value \(1230.0 \mathrm{~m}\) with the correct number of significant figures and appropriate Greek prefix? (a) \(1.23 \mathrm{~km}\) (b) \(1.230 \mathrm{~cm}\) (c) \(1.2300 \mathrm{~km}\) (d) \(1.2300 \mathrm{~mm}\) (e) \(12.3 \mathrm{~km}\)

Step-by-Step Solution

Answer

The correct choice is (c) 1.2300 km, as it has the correct number of significant figures (5) and the appropriate Greek prefix 'kilo' for the given measurement.

Step 1: Identify the number of significant figures in the given measurement

The given measurement is 1230.0 meters. The number 1230.0 contains 5 significant figures. The trailing zero after the decimal point is significant.

Step 2: Analyze the choices

Let's analyze each choice: (a) 1.23 km: This choice has only 3 significant figures, and the Greek prefix 'kilo' (km) means 'thousand', which is the correct range for the given measurement. (b) 1.230 cm: This choice has 4 significant figures but the Greek prefix 'centi' (cm) means 'hundredth', which is not the correct range for the given measurement. (c) 1.2300 km: This choice has 5 significant figures and 'kilo' (km) means 'thousand', which is the correct range. (d) 1.2300 mm: This choice has 5 significant figures but the Greek prefix 'milli' (mm) means 'thousandth', which is not the correct range for the given measurement. (e) 12.3 km: This choice has only 3 significant figures, and 'kilo' (km) means 'thousand', which is the correct range for the given measurement.

Step 3: Choose the correct option

Option (c) 1.2300 km is the correct choice, as it has the correct number of significant figures (5) and the appropriate Greek prefix 'kilo' for the given measurement.

Key Concepts

Greek Prefixes in Measurements

Understanding Greek prefixes in measurements is essential for science and engineering students. These prefixes are standardized indicators that represent specific multiples or fractions of a base unit in the International System of Units (SI). For example, the prefix 'kilo-' signifies a thousandfold of the base unit, while 'milli-' means a thousandth. A 'centi-' prefix indicates a hundredth of the base unit. These prefixes help simplify numerical representations and maintain precision without using large numbers of zeros.

For instance, when you see 1 kilometer or 1 km, it is understood that this is equivalent to 1,000 meters. Similarly, 1 milligram or 1 mg is 0.001 grams. Recognizing these prefixes is not just about memorization; it's about understanding the scale and context in which they're used, which can aid significantly in the measurement and conversion processes.

Measuring Value Precision

Measuring value precision refers to the exactness of a measurement. The concept of 'significant figures' is used to express precision in scientific notation. Significant figures, often shortened to 'sig figs', are the digits in a number that carry meaningful contributions to its measurement precision. This generally includes all digits except:

• Leading zeros, which serve only to place the decimal point.
• Trailing zeros when they are merely placeholders and do not indicate measurement precision.
• Spurious digits that exceed the precision of the measurement.

In our example with '1230.0 m', the value contains five significant figures. The trailing zero after the decimal point is significant because it indicates precision to the tenth of a meter. When converting or comparing measurements, maintaining the same level of precision is critical, as it reflects the reliability of the measurement.

Conversion of Units

Conversion of units is a basic scientific skill that involves changing from one unit of measurement to another without altering the quantity. To accurately convert units, you must use conversion factors that are precise ratios representing the relationship between the units. These factors are derived from the equivalence between different measurement systems or scales.

In our textbook problem, converting from meters to kilometers requires the knowledge that one kilometer equals one thousand meters. Using the conversion factor 1 km = 1000 m, we correctly transform '1230.0 m' into '1.2300 km', not altering the actual distance but simply representing it with a different, often more convenient unit of measurement. Proper conversion of units is indispensable for comparisons and computations in various scientific disciplines and real-life applications.