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Chapter 1: Problem 44

Indicate which of the following are exact numbers: (a) the mass of a 945-mL can of coffee, \((\mathbf{b})\) the number of students in your chemistry class, \((\mathbf{c})\) the temperature of the surface of the \(\operatorname{Sun},(\mathbf{d})\) the mass of a postage stamp, \((\mathbf{e})\) the number of milliliters in a cubic meter of water, \((\mathbf{f})\) the average height of \(\mathrm{NBA}\) basketball players.

Short Answer

Expert verified
Exact numbers: (b) number of students, (e) number of milliliters in cubic meter of water.

Step by step solution

01

Identify Fixed Quantities

Exact numbers are those that are known with complete precision, often because they arise from counting or are defined quantities in units. For example, the number of milliliters in a cubic meter of water is an exact number because it is a defined unit conversion: 1 cubic meter = 1,000,000 milliliters.
02

Evaluate Countable Quantities

Numbers that come from counting items are also exact because they are whole numbers without any estimation. The number of students in your chemistry class is exact since it is known precisely through counting.
03

Consider Measured Quantities with Uncertainty

Quantities like mass, temperature, and average height are not exact numbers because they typically involve measurement and are subject to some degree of uncertainty. For example, the mass of a 945-mL can of coffee, the temperature of the sun's surface, the mass of a postage stamp, and the average height of NBA players are estimated or measured with some precision but are not exact.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Unit conversion is a critical concept in science and mathematics, allowing us to express quantities in different units. This involves using specific conversion factors that define the relationships between units.
For example, the conversion from cubic meters to milliliters is a perfect illustration of this concept. One cubic meter is exactly 1,000,000 milliliters. There is no ambiguity here, as the conversion factor is a defined, exact number.

This precision is vital in scientific experiments and calculations, where consistency and accuracy are paramount. By relying on exact numbers for unit conversion, we reduce errors and ensure reliable results across various contexts.
  • Essential for scientific accuracy
  • Reduces errors in calculations
  • Provides exact relationships between units
Countable Quantities
Countable quantities refer to numbers that you acquire by directly counting distinct items. These quantities are precise because there is no room for ambiguity. You can simply count each item, and the result is an exact number.
For instance, if you have a chemistry class with 30 students, this number is exact because you can physically count each person present.

In everyday life, whenever you can point to individual items such as books on a shelf or chairs in a room, you are dealing with countable quantities. These figures are straightforward and perfect in terms of accuracy, as they are not influenced by external measurements or estimations.
  • Derived from direct counting
  • Exact and precise
  • No estimation involved
Measured Quantities
Measured quantities arise when we determine a property's value through a measuring instrument, and they are almost always associated with some level of uncertainty. This is because measurements can vary slightly depending on the instrument's precision and the person's skill in measuring.
For example, when you measure the mass of a postage stamp or the height of a basketball player, there鈥檚 always a degree of variability. The instruments used, such as scales or measuring tapes, have limits on how finely they can measure.

Thus, with measured quantities, you're noting an approximation that includes small, unavoidable errors, unlike countable quantities that are exact.
  • Have inherent uncertainty
  • Depend on instrument precision
  • Typically involve estimation
Uncertainty in Measurement
Uncertainty in measurement is an integral part of the scientific process. When measuring any physical quantity, there is always some level of uncertainty involved, as no measurement can be perfectly exact.
Measurements require tools, which can introduce minor errors based on limitations in precision and accuracy. Even external factors like temperature and environmental conditions can affect results.

Understanding and accounting for uncertainty in measurements is essential for interpreting data correctly. Scientists often express uncertainty with an error margin or confidence interval to gauge the reliability of their results.
  • Always present in measurements
  • Result of instrument limitations
  • Expressed as error margins

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Most popular questions from this chapter

A silvery metal is put inside a beaker of water. Bubbles form on the surface of the metal and it dissolves gradually. (a) Is this an example of a chemical or a physical change? (b) Do you expect the remaining solution to be a pure substance or a mixture?

(a) A sample of tetrachloroethylene, a liquid used in dry cleaning that is being phased out because of its potential to cause cancer, has a mass of \(40.55 \mathrm{~g}\) and a volume of \(25.0 \mathrm{~mL}\) at \(25^{\circ} \mathrm{C}\). What is its density at this temperature? Will tetrachloroethylene float on water? (Materials that are less dense than water will float.) (b) Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is a gas at room temperature and pressure. However, carbon dioxide can be put under pressure to become a "supercritical fluid" that is a much safer dry-cleaning agent than tetrachloroethylene. At a certain pressure, the density of supercritical \(\mathrm{CO}_{2}\) is \(0.469 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the mass of a 25.0-mL sample of supercritical \(\mathrm{CO}_{2}\) at this pressure?

A match is lit to light a candle. The following observations are made: (a) The candle burns. (b) Some wax melts. (c) Melted wax solidifies on the candleholder. (d) Soot (carbon) is produced by the burning of the match and the candle. Which of these occurrences are due to physical changes, and which are due to chemical changes?

Give the derived SI units for each of the following quantities in base SI units: (a) acceleration \(=\) distance \(/\) time \(^{2}\) (b) force \(=\) mass \(\times\) acceleration (c) work \(=\) force \(\times\) distance (d) pressure \(=\) force/area (e) power = work/time (f) velocity = distance/time (g) energy \(=\) mass \(\times(\text { velocity })^{2}\)

The total rate at which power is used by humans worldwide is approximately 15 TW (terawatts). The solar flux averaged over the sunlit half of Earth is \(680 \mathrm{~W} / \mathrm{m}^{2}\) (assuming no clouds). The area of Earth's disc as seen from the Sun is \(1.28 \times 10^{14} \mathrm{~m}^{2}\). The surface area of Earth is approximately 197,000,000 square miles. How much of Earth's surface would we need to cover with solar energy collectors to power the planet for use by all humans? Assume that the solar energy collectors can convert only \(10 \%\) of the available sunlight into useful power.
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