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Chemistry: Units and Measurement

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Unit 2: Module 2: Unit conversions and problem solving with metric measurement

About this unit, topic a: metric unit conversions.

• Metric system: units of weight (Opens a modal)
• Metric system: units of distance (Opens a modal)
• Metric system: units of volume (Opens a modal)
• Estimate mass (grams and kilograms) Get 3 of 4 questions to level up!
• Estimating length (mm, cm, m, km) Get 5 of 7 questions to level up!
• Estimate volume (milliliters and liters) Get 3 of 4 questions to level up!

Topic B: Application of metric unit conversions

• How to convert kg to mg and T to oz (Opens a modal)
• Convert liters to milliliters (Opens a modal)
• Converting metric units of length (Opens a modal)
• Multi-step unit conversion examples (metric) (Opens a modal)
• Convert to smaller units (g and kg) Get 3 of 4 questions to level up!
• Convert to smaller units (mL and L) Get 3 of 4 questions to level up!
• Convert to smaller units (mm, cm, m, & km) Get 5 of 7 questions to level up!
• Metric conversions word problems Get 3 of 4 questions to level up!

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2: Measurement and Problem Solving

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Chemistry, like all sciences, is quantitative. It concerns quantities, things that have amounts and units. Dealing with quantities and relating them to one another is very important in chemistry. In this chapter, we will discuss how we deal with numbers and units, including how they are combined and manipulated.

• 2.1: Taking Measurements Chemists measure the properties of matter and express these measurements as quantities. A quantity is an amount of something and consists of a number and a unit. The number tells us how many (or how much), and the unit tells us what the scale of measurement is. For example, when a distance is reported as “5 kilometers,” we know that the quantity has been expressed in units of kilometers and that the number of kilometers is 5.
• 2.2: Scientific Notation - Writing Large and Small Numbers Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator requires a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes.
• 2.3: Significant Figures - Writing Numbers to Reflect Precision Uncertainty exists in all measurements. The degree of uncertainty is affected in part by the quality of the measuring tool. Significant figures give an indication of the certainty of a measurement. Rules allow decisions to be made about how many digits to use in any given situation.
• 2.4: Significant Figures in Calculations To round a number, first decide how many significant figures the number should have. Once you know that, round to that many digits, starting from the left. If the number immediately to the right of the last significant digit is less than 5, it is dropped and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1.
• 2.5: The Basic Units of Measurement Metric prefixes derive from Latin or Greek terms. The prefixes are used to make the units manageable. The SI system is based on multiples of ten. There are seven basic units in the SI system. Five of these units are commonly used in chemistry.
• 2.6: Problem Solving and Unit Conversions During your studies of chemistry (and physics also), you will note that mathematical equations are used in a number of different applications. Many of these equations have a number of different variables with which you will need to work. Note also that these equations will often require the use of measurements with their units. Algebra skills become very important here!
• 2.7: Solving Multi-step Conversion Problems Sometimes you will have to perform more than one conversion to obtain the desired unit.
• 2.8: Units Raised to a Power Conversion factors for area and volume can also be produced by the dimensional analysis method. Remember that if a quantity is raised to a power of 10, both the number and the unit must be raised to the same power of 10.
• 2.9: Density Density is a physical property found by dividing the mass of an object by its volume. Regardless of the sample size, density is always constant.
• 2.E: Measurement and Problem Solving (Exercises) Exercises for Chapter 2 of Tro's Introductory Chemistry textmap.

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Chapter 1: Algebra Review

1.6 Unit Conversion Word Problems

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

Example 1.6.1

 If 1 pound = 16 ounces, how many pounds are in 435 ounces?

[latex]\begin{array}{rrll} 435\text{ oz}&=&435\cancel{\text{oz}}\times \dfrac{1\text{ lb}}{16\cancel{ \text{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}[/latex]

The same process can be used to convert problems with several units in them. Consider the following example.

Example 1.6.2

A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?

[latex]\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{45\cancel{\text{mi}}}{\cancel{\text{hr}}}\times \dfrac{5280 \text{ ft}}{1\cancel{ \text{mi}}}\times \dfrac{1\cancel{\text{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}[/latex]

Example 1.6.3

Convert 8 ft 3 to yd 3 .

[latex]\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}[/latex]

Example 1.6.4

A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft 2 (area = length × width).

Converting the area yields:

[latex]\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}[/latex]

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

Example 1.6.5

A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?

[latex]\begin{array}{rrll} 3\text{ prescriptions}&=&3\cancel{\text{pres.}}\times \dfrac{60\cancel{\text{tablets}}}{1\cancel{\text{pres.}}}\times \dfrac{4\cancel{\text{mg}}}{1\cancel{\text{tablet}}}\times \dfrac{1\text{ dosage}}{12\cancel{\text{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}[/latex]

Metric and Imperial (U.S.) Conversions

[latex]\begin{array}{rrlrrl} 12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\ 3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\ 1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\ 5280\text{ ft}&=&1\text{ mi}&&& \end{array}[/latex]

Imperial to metric conversions:

[latex]\begin{array}{rrl} 1\text{ inch}&=&2.54\text{ cm} \\ 1\text{ ft}&=&0.3048\text{ m} \\ 1\text{ mile}&=&1.61\text{ km} \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\ 43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\ 640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2 \end{array}[/latex]

[latex]\begin{array}{rrl} 1\text{ in}^2&=&6.45\text{ cm}^2 \\ 1\text{ ft}^2&=&0.092903\text{ m}^2 \\ 1\text{ mi}^2&=&2.59\text{ km}^2 \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\ 4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\ 42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3 \end{array}[/latex]

[latex]\begin{array}{rrl} 16.39\text{ cm}^3&=&1\text{ in}^3 \\ 1\text{ ft}^3&=&0.0283168\text{ m}^3 \\ 3.79\text{ litres}&=&1\text{ gal} \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\ 16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\ 2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton} \end{array}[/latex]

[latex]\begin{array}{rrl} 453\text{ g}&=&1\text{ lb} \\ 2.2\text{ lb}&=&1\text{ kg} \end{array}[/latex]

Temperature

Fahrenheit to Celsius conversions:

[latex]\begin{array}{rrl} ^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} - 32) \\ \\ ^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32) \end{array}[/latex]

For questions 1 to 18, use dimensional analysis to perform the indicated conversions.

• 7 miles to yards
• 234 oz to tons
• 11.2 mg to grams
• 1.35 km to centimetres
• 9,800,000 mm to miles
• 4.5 ft 2 to square yards
• 435,000 m 2 to square kilometres
• 8 km 2 to square feet
• 0.0065 km 3 to cubic metres
• 14.62 in 3 to square centimetres
• 5500 cm 3 to cubic yards
• 3.5 mph (miles per hour) to feet per second
• 185 yd per min. to miles per hour
• 153 ft/s (feet per second) to miles per hour
• 248 mph to metres per second
• 186,000 mph to kilometres per year
• 7.50 tons/yd 2 to pounds per square inch
• 16 ft/s 2 to kilometres per hour squared

For questions 19 to 27, solve each conversion word problem.

• On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
• A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
• An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
• Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
• You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
• A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
• A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a [latex]\frac{1}{3}[/latex]-acre lot (1 acre = 43,560 ft 2 ). What is the maximum allowed footprint for your house in square feet? In square metres?
• A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
• The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?

Answer Key 1.6

Intermediate Algebra by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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