Problem 108 - General Chemistry: Principles and Modern Applications

Question

List the following in the order of increasing precision, indicating any quantities about which the precision is uncertain: (a) \(1400 \mathrm{km} ;\) (b) \(1516 \mathrm{kg} ;\) (c) \(0.00304 \mathrm{g};\) (d) \(125.34 \mathrm{cm} ;\) (e) \(2000 \mathrm{mg}\).

Step-by-Step Solution

Answer

The measurements ranked in order of increasing precision are: 1400 km, 1516 kg, 2000 mg, 125.34 cm, and 0.00304 g. The precision remains uncertain for 1400 km and 2000 mg because they end with zero.

Step 1: Understand the Concept of Precision

Precision in measurements refers to the smallest possible unit or increment. A measurement with a smaller unit is considered more precise.

Step 2: Identify Precision for Each Quantity

In order to rank the quantities, one must look at each given measure and identify its degree of precision. Here are the identified levels: (a) 1400 km is precise to the nearest 100 km, (b) 1516 kg is precise to the nearest 1 kg, (c) 0.00304 g is precise to the nearest 0.00001 g, (d) 125.34 cm is precise to the nearest 0.01 cm, and (e) 2000 mg is precise to the nearest 1 mg.

Step 3: Understand Uncertainty in Measurements

Uncertainty generally arises when a measure ends with 0. It is unclear whether that zero is a part of a precise measurement or it was put on approximated basis. In this set, (a) 1400 km and (e) 2000 mg have an uncertainty because they end with zero.

Step 4: Organise the Measures in Order of Increasing Precision

After identifying the degree of precision for each, they can be ordered from the least precise to the most: (a) 1400 km, (b) 1516 kg, (e) 2000 mg, (d) 125.34 cm, and (c) 0.00304 g. This order represents increasing precision.

Key Concepts

Measurement Uncertainty

Measurement uncertainty is an intrinsic aspect of any measurement process. It refers to the doubt that exists about the result of any measurement. This uncertainty can come from various sources:

• Instrument limitations: Every measuring device has limitations based on its design. For instance, a ruler might measure to the nearest millimeter, but it cannot account for fractions of a millimeter.
• Human error: Misreading an instrument or making a mistake in recording data can introduce errors.
• Environmental influences: Changes in temperature, humidity, or other conditions can affect the measurement.

When a measurement ends with zeroes, uncertainty can be particularly pronounced. For example, in the given exercise, 1400 km and 2000 mg end in zeros, which makes it unclear if they were rounded or measured exactly. Due to this ambiguity, it becomes a challenge to represent just how precise these measurements are.

Understanding uncertainty allows us to interpret the reliability and accuracy of the data we work with, enabling better decision-making in scientific and everyday contexts.

Precision Ranking

Precision ranking involves sorting measurements based on their degree of exactness. Precision refers to how finely a measurement can be made. This is different from accuracy, which concerns how close a measurement is to the true value.

In the exercise provided, we determine precision by considering the smallest unit of each measurement. To rank the measurements by increasing precision, we consider the units of measurement and their respective increments:

• ((a) 1400 km - precise to the nearest 100 km)
• ((b) 1516 kg - precise to the nearest 1 kg)
• ((e) 2000 mg - precise to the nearest 1 mg)
• ((d) 125.34 cm - precise to the nearest 0.01 cm)
• ((c) 0.00304 g - precise to the nearest 0.00001 g)

Thus, the precision of these measurements increases from 1400 km to 0.00304 g.

Understanding this hierarchy helps ensure that the right tools and units are used for different scientific experiments or practical tasks, guaranteeing an appropriate level of detail and statistical analysis.

Units and Precision

Units are fundamental in conveying precision, as they represent the scale and increments used during measurement. Every measurement unit offers varying levels of precision, which defines how specific a measurement can be.

For instance, in length, units like kilometers (km) suggest less precision due to larger increments, whereas centimeters (cm) offer more precision. In the exercise we analyzed,

• The measurement "1400 km" highlights larger increments, making it less precise.
• While "0.00304 g" uses a very small unit of grams, hence it is more precise.

How precisely we can state a measurement depends on both the unit itself and the smallest subdivision of that unit. Using appropriate units is vital in controlling and improving the precision of any measurement.

By selecting proper units based on required precision, scientists and practitioners ensure that measurements not only fit their purpose but also reduce the potential for error, leading to more accurate and reliable data.