Question

Indicate which of the following are exact numbers: (a) the mass of a 32 -oz can of coffee, \((b)\) the number of students in your chemistry class, (c) the temperature of the surface of the sun, (d) the mass of a postage stamp, (e) the number of milliliters in a cubic meter of water, (f) the average height of students in your school.

Short answer

The exact numbers among the given quantities are (b) the number of students in your chemistry class and (e) the number of milliliters in a cubic meter of water. The other quantities involve measurements, approximations, or uncertainties and therefore are not considered exact numbers.

Step-by-step solution

(a) Mass of a 32-oz can of coffee:

The mass of a 32-oz can of coffee is not an exact number because the mass of the coffee can vary depending on various factors such as packaging and storage conditions. Moreover, the conversion of ounces to grams will not be exact either due to measurement uncertainties.

(b) Number of students in your chemistry class:

The number of students in a chemistry class is an exact number, as it is a specific, whole number that can be counted with certainty. There are no approximations or uncertainties involved in counting students.

(c) Temperature of the surface of the sun:

The temperature of the surface of the sun is not an exact number. It is a measurement taken from scientific observations and is subject to uncertainties and potential changes over time.

(d) Mass of a postage stamp:

The mass of a postage stamp is not an exact number, as it can vary depending on the paper used, ink, and other slight manufacturing variations. Moreover, even if we know the nominal mass of a postage stamp, it might not be the same for all stamps due to those manufacturing variations.

(e) Number of milliliters in a cubic meter of water:

The number of milliliters in a cubic meter of water is an exact number because there is a precise and definite conversion from cubic meters to milliliters: 1 cubic meter equals \(1,000,000\) milliliters. There are no uncertainties or approximations involved in this conversion.

(f) Average height of students in your school:

The average height of students in your school is not an exact number. It is an average based on measurements of individual students, which are subject to uncertainties and approximations. The average can also change as new students enroll or current students grow.

Key concepts

Measurement Uncertainty

Understanding measurement uncertainty is crucial in the field of chemistry, as it pertains to the reliability and precision of data obtained through various measurements. Essentially, measurement uncertainty acknowledges that no measurement can be absolutely perfect due to limitations in measurement instruments, human error, and other contributing factors.

For example, when weighing a can of coffee, a scale might provide a reading, but that reading can be slightly off due to calibration issues, changes in air pressure, or temperature variations. The same concept applies to other measurements like the temperature of the sun or the mass of a postage stamp; external factors always introduce a small degree of uncertainty.

Even when numbers seem precise - such as the conversion of mass units or liquid volumes - the underlying issue is not the mathematics but the initial measurement. Remember, quantified uncertainties are integral in scientific communication, as they give a range within which the true value likely exists. When a number is described as 'exact' in chemistry, it usually refers to a value that is defined or counted with certainty, not dependent on a measurement tool.

Scientific Observation

Scientific observation is the process of collecting information about the physical world using our senses or tools designed to extend our senses and provide data. In chemistry and other sciences, observations need to be accurate and precise to be valuable, but they often come with inherent uncertainty.

Considering the sun's surface temperature, scientists must rely on specialized tools and inferred data since direct measurement isn't possible. These indirect observations may include sophisticated instruments that detect radiation or other emissions from the sun, leading to calculations that estimate temperature. This process exemplifies how scientific observations strive for accuracy but must always account for some level of uncertainty.

Improving Observational Accuracy

Researchers work to enhance accuracy and precision using calibration, repeated trials, and refined instrument technology. Each improvement aims to reduce the potential for error in scientific observations, providing a better understanding of the observed phenomena.

Unit Conversion

Unit conversion in chemistry is a critical skill that enables scientists to communicate measurements effectively and compare data. It involves changing a quantity expressed in one set of units to another set of units without altering the value itself.

For instance, converting milliliters to cubic meters is a common conversion in chemistry that involves mathematical precision. The exactness of such conversions relies on the defined relationship between units: one cubic meter is exactly equivalent to one million milliliters. Gaining fluency in these conversions allows for seamless transition between systems of measurement, such as metric to imperial, which is necessary when working in international or interdisciplinary contexts.

Unit conversion accuracy is essential not only for achieving correct results in chemical experiments and calculations but also for ensuring consistency and understanding across different scientific communities and publications.