Problem 157 - Introductory Chemistry Atoms First

Question

Indicate the uncertainty in: (a) \(74.8 \mathrm{~m}\) (b) \(0.0026 \mathrm{~g}\) (c) \(1.250 \times 10^{3} \mathrm{~L}\) (d) \(18 \mathrm{~cm}\) (e) 18 pennies

Step-by-Step Solution

Answer

(a) \(74.8 \pm 0.1 \mathrm{~m}\) (b) \(0.0026 \pm 0.0001 \mathrm{~g}\) (c) \((1.250 \pm 0.010) \times 10^{3} \mathrm{~L}\) (d) \(18 \pm 1 \mathrm{~cm}\) (e) 18 pennies

Step 1: Calculate Uncertainty for (a)

For the measurement \(74.8\mathrm{~m}\), the least significant digit is in the tenths place. Thus, the uncertainty in this measurement is represented as ±0.1 m. The measurement with its uncertainty can be written as \(74.8 \pm 0.1 \mathrm{~m}\).

Step 2: Calculate Uncertainty for (b)

For the measurement \(0.0026\mathrm{~g}\), the least significant digit is in the ten-thousandths place. Therefore, the uncertainty in this measurement is represented as ±0.0001 g. The measurement with its uncertainty can be written as \(0.0026 \pm 0.0001 \mathrm{~g}\).

Step 3: Calculate Uncertainty for (c)

For the measurement \(1.250 \times 10^{3}\mathrm{~L}\), the least significant digit is in the hundredths place. Thus, the uncertainty in this measurement is represented as ±0.01 L. However, the measurement is expressed in scientific notation, so we should express the uncertainty in the same format. The measurement with its uncertainty can be written as \((1.250 \pm 0.010) \times 10^{3} \mathrm{~L}\).

Step 4: Calculate Uncertainty for (d)

For the measurement \(18\mathrm{~cm}\), the least significant digit is in the ones place. Thus, the uncertainty in this measurement is represented as ±1 cm. The measurement with its uncertainty can be written as \(18 \pm 1 \mathrm{~cm}\).

Step 5: Calculate Uncertainty for (e)

The uncertainty associated with counting something, such as 18 pennies, is not due to the measuring instrument. Instead, this is a discrete counting measurement with no uncertainty, since we're sure of the number of pennies counted. Therefore, the measurement can be written as 18 pennies with no associated uncertainty.

Key Concepts

Understanding Significant Figures

Significant figures are crucial in expressing the precision of measurements in science. They refer to the digits in a number that are known with certainty plus the first uncertain digit. For instance, in the measurement of \(74.8\, \text{m}\), the number has three significant figures: '7', '4', and '8'. The digit '8' is the least significant because it determines the precision of the measurement, indicating the uncertainty. There are several rules to identify significant figures:

• All nonzero digits are significant.
• Any zeros between significant digits are also significant.
• Leading zeros, such as in \(0.0026\, \text{g}\), are not counted as significant.
• Trailing zeros in a number with a decimal point are significant.

Remember, the purpose of significant figures is to communicate the precision of your measurements clearly. It ensures that the level of accuracy is maintained consistently throughout calculations.

The Power of Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently, using powers of ten. This method simplifies calculations and ensures uniformity in the representation of numbers, especially in scientific work. For example, the measurement \(1.250 \times 10^3\, \text{L}\) represents the number 1250 with three significant figures, clearly showing the precision of the measurement. The format is defined as a product of a number between 1 and 10 and a power of 10. Here’s how it works:

• For numbers greater than one, move the decimal place to the left until there's only one non-zero digit in front of it, adjusting the power of ten accordingly.
• For numbers less than one, such as \(0.0026\, \text{g}\), move the decimal to the right, with the power of ten reflecting the number of places you moved.

Scientific notation not only makes numbers easier to read and write but also indicates the significant figures, reflecting the true precision of the measurement.

Precision in Measurement

Precision in measurement refers to how repeatable or consistent a measurement is. It is reflected by the number of significant figures and often expressed through measurement uncertainty, such as an error margin. A precise measurement is one with minimal variations among readings when repeated under unchanged conditions. For example, when a length is measured as \(74.8 \pm 0.1\, \text{m}\), the precision is high because the uncertainty is small. Conversely, \(18 \pm 1\, \text{cm}\) shows lower precision due to its larger uncertainty. Factors influencing precision include:

• The equipment used - more sophisticated instruments typically allow for more precision.
• The skill of the person making the measurement - experienced operators can achieve greater consistency.

Ultimately, precision is about consistency among multiple measurements, and helps convey the reliability of your data.