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Chapter 1: Problem 58

How many significant figures are there in each of the following measurements? a) \(4.0100 \mathrm{mg}\) b) \(0.05930 \mathrm{~g}\) c) \(0.035 \mathrm{~mm}\) d) \(3.100 \mathrm{~s}\) e) \(8.91 \times 10^{1} \mathrm{~L}\) f) \(9.100 \times 10^{4} \mathrm{~cm}\)

Short Answer

Expert verified
a) 5, b) 4, c) 2, d) 4, e) 3, f) 4 significant figures.

Step by step solution

01

Identify Significant Figures Rule 1

Any non-zero digits in a measurement are always considered significant. For example, in the number 8.91, the digits '8' and '9' are non-zero and therefore significant.
02

Identify Significant Figures Rule 2

Any zeros between non-zero digits are considered significant. For instance, in the number 9.100, the zero between '9' and '1' is significant.
03

Identify Significant Figures Rule 3

Leading zeros, which precede all non-zero digits and are found in decimal numbers, are not significant. For example, in 0.035 the zeros before '3' are not significant.
04

Identify Significant Figures Rule 4

Trailing zeros in a number with a decimal point are significant. In 4.0100, the zeros after the '1' are significant because they are at the end of the number after the decimal point.
05

Apply Rules to Determine A

For 4.0100 mg, the digits '4', '0', '1', '0', '0' are significant. All zeros are important here because they are either between non-zero digits or trailing in a decimal. Hence, there are 5 significant figures.
06

Apply Rules to Determine B

For 0.05930 g, the digits '5', '9', '3', '0' are significant. The leading zeros are not counted as significant, leaving us with 4 significant figures.
07

Apply Rules to Determine C

For 0.035 mm, only the digits '3' and '5' are significant. The leading zeros are not counted, so there are 2 significant figures.
08

Apply Rules to Determine D

For 3.100 s, the digits '3', '1', '0', '0' are significant. The trailing zeros count because they come after a decimal point, resulting in 4 significant figures.
09

Apply Rules to Determine E

For 8.91 脳 10^1 L, the scientific notation does not affect significant figures. The digits '8', '9', '1' are significant, resulting in 3 significant figures.
10

Apply Rules to Determine F

For 9.100 脳 10^4 cm, the digits '9', '1', '0', '0' are significant, due to the presence of the decimal point. This gives 4 significant figures.

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Key Concepts

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

Scientific Notation
Scientific notation is a way to express very large or very small numbers in a concise manner.
It is written as the product of a number (usually between 1 and 10) and a power of ten.
This method is particularly useful in scientific fields where measurements can vary widely in scale. For example, the number 0.00045 can be written as \(4.5 \times 10^{-4}\).
Some key points about scientific notation include:
  • It allows for easier readability of unwieldy numbers.
  • Significant figures in scientific notation are determined by the significant digits in the coefficient (the number before the \(\times 10^n\)).
  • The power of ten does not influence the number of significant figures.
When dealing with scientific notation, it's important to focus on the digits in the coefficient to determine precision and significance.
Measurement Precision
Measurement precision refers to the level of detail in a measurement. It is represented by the amount of significant figures that are used.
These figures reflect the certainty or reliability of the measurement. There are several rules to identify significant figures which guide precision:
  • Any non-zero digits are always considered significant.
  • Zeros between non-zero digits are significant.
  • Leading zeros, or zeros that precede the first non-zero digit, are not significant.
  • Trailing zeros, which appear after a decimal point and at the end of a number, are significant.
With these rules, one can determine how precise a measurement is. High precision means more significant figures, which implies greater detail and accuracy in the measurement.
Decimal Numbers
Decimal numbers are numbers that contain a decimal point, allowing for fractional values.
They are used to express numbers with precision beyond whole numbers. In the context of significant figures, the placement of zeros in decimal numbers is critical:
  • Leading zeros in decimal numbers are not significant, as they simply position the decimal point.
  • Trailing zeros in decimal numbers, however, emphasize measurement precision and therefore are significant.
  • Decimal points themselves indicate that all digits, starting with the leftmost non-zero digit, are significant.
This nuanced understanding helps in identifying the meaningful figures in measurement. For instance, in the number 0.05930, only the digits 5, 9, 3, and the zero after 3 are significant, reflecting a precision up to the thousandths place.

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Most popular questions from this chapter

When chlorine gas is bubbled into a solution of sodium bromide, the sodium bromide reacts to give bromine, a red-brown liquid, and sodium chloride (ordinary table salt). A solution was made by dissolving \(20.6 \mathrm{~g}\) of sodium bromide in \(100.0 \mathrm{~g}\) of water. After passing chlorine through the solution, investigators analyzed the mixture. It contained \(16.0 \mathrm{~g}\) of bromine and \(11.7 \mathrm{~g}\) of sodium chloride. How many grams of chlorine reacted?

For each of the following, write the measurement in terms of an appropriate prefix and base unit. A The mass of calcium per milliliter in a sample of blood serum is \(0.0912 \mathrm{~g}\). B The radius of an oxygen atom is about \(0.000000000066 \mathrm{~m}\). C A particular red blood cell measures \(0.0000071 \mathrm{~m}\). D The wavelength of a certain ultraviolet radiation is \(0.000000056 \mathrm{~m} .\)

Titanium metal is used in aerospace alloys to add strength and corrosion resistance. Titanium melts at \(1677^{\circ} \mathrm{C}\). What is this temperature in degrees Fahrenheit?

Describe each of the following as a physical or chemical property of each listed chemical substance. A chlorine is a green gas B zinc reacts with acids C magnesium has a density of \(1.74 \mathrm{~g} / \mathrm{cm}^{3}\) D iron can rust

You have a piece of gold jewelry weighing \(9.35 \mathrm{~g}\). Its volume is \(0.654 \mathrm{~cm}^{3}\). Assume that the metal is an alloy (mixture) of gold and silver, which have densities of \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\) and \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. Also assume that there is no change in volume when the pure metals are mixed. Calculate the percentage of gold (by mass) in the alloy. The relative amount of gold in an alloy is measured in karats. Pure gold is 24 karats; an alloy of \(50 \%\) gold is 12 karats. State the proportion of gold in the jewelry in karats.
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