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Chapter 2: Problem 35

Express each number in decimal notation (i.e., express the number without using scientific notation). (a) \(6.022 \times 10^{23}\) (number of carbon atoms in \(12.01 \mathrm{~g}\) of carbon) (b) \(1.6 \times 10^{-19} \mathrm{C}\) (charge of a proton in coulombs) (c) \(2.99 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (speed of light) (d) \(3.44 \times 10^{2} \mathrm{~m} / \mathrm{s}\) (speed of sound)

Short Answer

Expert verified
(a) 602,200,000,000,000,000,000,000 (b) 0.00000000000000000016 (c) 299,000,000 (d) 344

Step by step solution

01

- Convert Scientific Notation to Decimal Notation for 6.022 x 10^23

To express the number 6.022 x 10^23 in decimal notation, move the decimal point 23 places to the right as we have a positive exponent. Fill any empty positions with zeros.
02

- Convert Scientific Notation to Decimal Notation for 1.6 x 10^-19

To express the number 1.6 x 10^-19 in decimal notation, move the decimal point 19 places to the left as we have a negative exponent. Fill any empty positions with zeros.
03

- Convert Scientific Notation to Decimal Notation for 2.99 x 10^8

To express the number 2.99 x 10^8 in decimal notation, move the decimal point 8 places to the right. Fill any empty positions with zeros.
04

- Convert Scientific Notation to Decimal Notation for 3.44 x 10^2

To express the number 3.44 x 10^2 in decimal notation, move the decimal point 2 places to the right as indicated by the exponent.

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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 concise way to express very large or very small numbers, which is commonly used in fields like chemistry and physics.
It consists of a decimal part called the 'coefficient', which is typically a number between 1 and 10, and an 'exponent', which shows how many places to shift the decimal point. For example, in the scientific notation \(6.022 \times 10^{23}\), the coefficient is 6.022 and the exponent is 23.
This notation greatly simplifies calculations, as it reduces the number of zeros needed to write out the number fully.
Decimal Notation
Decimal notation is the standard form of writing numbers using the base-10 number system. It's the way we typically write numbers, with each position indicating a power of 10. When converting from scientific to decimal notation, you'll shift the decimal point right for positive exponents and left for negative exponents until the number is displayed with no powers of ten. This form is more intuitive for everyday use but can become unwieldy with very large or small numbers.
Exponents
Exponents are used to indicate how many times a number, the base, is to be multiplied by itself. In scientific notation, a positive exponent means you move the decimal to the right, and a negative exponent means you move it to the left. For instance, \(10^{3}\) means \(10 \times 10 \times 10 = 1000\), and \(10^{-2}\) is \(0.01\), because you move the decimal place two positions to the left from 1.
Chemistry Calculations
Chemistry calculations often involve handling very large or very small numbers, like counting the number of molecules in a gram of a substance or measuring reaction rates. Here, scientific notation becomes particularly useful to communicate and compute these quantities effectively.
In the exercise, the first number represented the number of carbon atoms in 12.01 grams of carbon, which is a fundamental application in chemistry where precision and ability to handle large numbers are crucial.
Avogadro's Number
Avogadro's number, \(6.022 \times 10^{23}\), represents the number of atoms in one mole of a substance and is a cornerstone of chemistry for relating macroscopic amounts of substances to the number of particles. Understanding how to translate this large number from scientific notation to decimal is important for visualizing and measuring actual amounts of substances in laboratory settings.
Proton Charge
The charge of a proton is approximately \(1.6 \times 10^{-19}\) coulombs, a fundamental physical constant. The small magnitude of this charge is best expressed in scientific notation, but learning to convert it to decimal notation helps in understanding electricity at the molecular and atomic levels, as well as in calculations involving electric forces and fields in physics.
Speed of Light
The speed of light in a vacuum is approximately \(2.99 \times 10^{8}\) meters per second, a fundamental constant in physics. This speed is central to the theory of relativity and various forms of astronomical and physical measurements. Although it's usually expressed in scientific notation, being able to convert it to decimal form may help with understanding the vast scales of space and speeds in the universe.
Speed of Sound
The speed of sound in air at room temperature is roughly \(3.44 \times 10^{2}\) meters per second. Unlike the speed of light, the speed of sound is much smaller and often expressed in decimal notation without difficulty. Understanding this conversion is essential in studies related to acoustics, engineering, and understanding the principles of wave propagation.

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Most popular questions from this chapter

Express each number in scientific notation. (a) \(38,802,000\) (population of California) (b) \(1,419,000\) (population of Hawaii) (c) \(19,746,000\) (population of New York) (d) 584,000 (population of Wyoming)

An aluminum engine block has a volume of \(4.77 \mathrm{~L}\) and a mass of \(12.88 \mathrm{~kg}\). What is the density of the aluminum in grams per cubic centimeter?

Perform each calculation to the correct number of significant figures. (a) \(89.3 \times 77.0 \times 0.08\) (b) \(\left(5.01 \times 10^{5}\right) \div\left(7.8 \times 10^{2}\right)\) (c) \(4.005 \times 74 \times 0.007\) (d) \(453 \div 2.031\)

The average U.S. farm occupies 435 acres. How many square miles is this? \(\left(1\right.\) acre \(\left.=43,560 \mathrm{ft}^{2} ; 1 \mathrm{mi}=5280 \mathrm{ft}\right)\)

A supposedly gold crown is tested to determine its density. It displaces \(10.7 \mathrm{~mL}\) of water and has a mass of \(206 \mathrm{~g}\). Could the crown be made of gold?
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