Conversion Factors And Dimensional Analysis
A ratio of two equivalent quantities expressed with different measurement units can be used as a unit conversion factor. For example, the lengths of 2.54 cm and 1 in. are equivalent , and so a unit conversion factor may be derived from the ratio,
\large\text\text\frac}}\text\text\frac}}
Several other commonly used conversion factors are given in Table 1.
Table 1. Common Conversion Factors | |
---|---|
Length | |
1 mi = 5280 ft | 1 gal = 3.785 L |
When we multiply a quantity by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units . For example, a basketball players vertical jump of 34 inches can be converted to centimeters by:
\large\text\cancel}\times \frac}}}=\text
Since this simple arithmetic involves quantities, the premise of dimensional analysis requires that we multiply both numbers and units. The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield \large\frac\times \text}} . Just as for numbers, a ratio of identical units is also numerically equal to one, \large\frac}}=\text and the unit product thus simplifies to cm. Using dimensional analysis, we can determine that a unit conversion factor has been set up correctly by checking to confirm that the original unit will cancel, and the result will contain the sought unit.
Metric To Metric Conversions
How Many Kilograms Are in 1,532 Grams?
The graphic shows seven steps to convert grams to kilograms.Step A shows the relationship between kilograms and grams.
In Step B, both sides of the equation are divided by 1000 g.
Step C shows how the value of 1 kg/1000 g is the equal to the number 1. This step is important in the unit cancellation method. When you multiply a number or variable by 1, the value is unchanged.
Step D restates the example problem.
In Step E, multiply both sides of the equation by 1 and substitute the left side’s 1 with the value in step C.
Step F is the unit cancellation step. The gram unit from the top of the fraction is canceled from the bottom leaving only the kilogram unit.
Dividing 1536 by 1000 yields the final answer in step G.
The final answer is: There are 1.536 kg in 1536 grams.
Example : Square Feet To Square Meters
Sometimes you have to deal with squared units. In the US, you oftensee them with a sq prefix. But they are actually easier tomanipulate if you treat them just like variables and use the² sign.
I correspond with a friend outside the US, and we are describingour homes to each other. If my apartment measures 850 square feet, whatis that in square meters? In other words, convert 850 ft² tom².
Solution: I need a fraction equal to 1, with m² on the top andft² on the bottom. The way to obtain that is to form a fractionequal to 1 with plain m on the top and plain ft on the bottom, andthen square it .
As it happens, I dont remember the conversion from feet to meters, but I do remember the conversions between both of them and inches: |
1 ft = 12.00 in1 m = 39.37 in |
So I construct my fraction in two steps: |
1 = 1 × 1 1 m 12.00 in1 = -------- × -------- 39.37 in 1 ft 12.00 m1 = -------- 39.37 ft 0.3048 m1 = -------- ft |
Now remember that the original measurement is in ft². Therefore I must multiply the original measurement, 850 ft², by the square of the above fraction, to get ft² in the denominator and match the ft² in the original measurement: |
²850 ft² × |
When a fraction is squared, thats the same as squaring the top and squaring the bottom, including units: |
850 × 0.3048² ft² m² -------------------- ft² |
Divide through by ft² top and bottom, and do the arithmetic to get the answer: |
850 ft² = 79 m² |
You May Like: What Is Running Water In Geography
Significant Digits In Conversions
English-English and metric-metric conversions are always exact. In other words, when converting from one unit to another unit in the same system , the conversions have been defined and are not measurements so they do not affect the significant digits of the calculation.
1 ft = 12 in. 1 yd = 3 ft 1 gal =4 qt
1 km = 103 m 1 cg = 10-2 g 1 GL = 109 L
However, English-metric conversions are inexact. They have been measured and not defined so the significant digits do affect the significant digits of the calculation.
1 lb 454 g 1 qt 0.9463 L
The only exact English-metric conversion is 1 in. = 2.54 cm which was defined in 1958. This conversion will not affect the significant digits of the calculation.
What Will You Learn And What You Need To Do A Organic Conversion
You can check your knowledge about organic compounds and their reactions from organic conversions exercises. Also organic conversions will help you to get remembered those organic compounds and their reactions again again again.
In this tutorial, we discuss about
Please enable JavaScript
Recommended Reading: Algebra 2 Chapter 3 Test Answer Key
Problem Solving With Multiple Conversions
Sometimes you will have to perform more than one conversion to obtain the desired unit. For example, suppose you want to convert 54.7 km into millimeters. You can either memorize the relationship between kilometers and millimeters, or you can do the conversion in two steps. Most people prefer to convert in steps.
To do a stepwise conversion, we first convert the given amount to the base unit. In this example, the base unit is meters. We know that there are 1,000 m in 1 km:
Then we take the result and convert it to millimeters, remembering that there are \ for every \:
\ & = 5.47 \times 10^7\ \rm \end \nonumber \]
We have expressed the final answer in scientific notation.
As a shortcut, both steps in the conversion can be combined into a single, multistep expression:
Worked Examples Of Converting Joules To Kilojoules And Kilojoules To Joules
Question 1: Convert 1 kilojoule to joules
Solution:
From the table above we see that kilo = 103 = 1,000
1 kJ = 103 J = 1,000 J
1 kilojoule = 1,000 joules
Question 2: Convert 2.5 kJ to joules
Solution:
From the table above we see that kilo = 103 = 1,000
1 kJ = 1,000 J
Multiply both sides of equation by 2.5:
2.5 × 1 kJ = 2.5 × 1,000 J
2.5 kJ = 2,500 J
Which we can express in scientific notation as:
2,500 J = 2.5 × 103 J
Question 3: Convert 5 millijoules to joules
Solution:
From the table above we see that milli = 10-3
1 mJ = 10-3 J
Multiply both sides of the equation by 5:
5 × 1 mJ = 5 × 10-3 J = 0.005 J
5 mJ = 0.005 J
Question 4: Convert 250 J to kilojoules
Solution:
From the table above we see that:
103 J = 1 kJ
Divide both sides of the equation by 1000 to find the number of kilojoules in 1 joule:
1 J = 10-3 kJ = 0.001 kJ
Multiply both sides of the equation by 250:
250 × 1 J = 250 × 0.001 kJ
250 J = 0.250 kJ
Question 5: Convert 25 µJ to kilojoules
Solution:
From the table above we see tha µ = 10-6
1 µJ = 10-6 J
Multiply both sides of this equation by 25 to determine the number of joules in 25 µJ :
25 × 1 µJ = 25 × 10-6 J
25 µJ = 2.5 × 10-5 J
Convert 2.5 × 10-5 J to kilojoules:
From the table above we see that:
1 kJ = 103 J
Divide both sides of this equation by 103:
1 kJ ÷ 103 = 1 J ÷ 103
10-3 kJ = 1 J
Multiply both sides of the equation by 2.5 × 10-5 to find the number of kJ in 2.5 × 10-5 J
× 10-3 kJ = × 1 J
2.5 × 10-8 kJ = 2.5 × 10-5 J
State the answer to the question:
Also Check: What Is Meant By Geography
Where To Find Conversion Factors
I dont just pull the conversion factors out of my hat. Many bookshave tables of conversions, including the venerable Handbook ofChemistry and Physics.
There are also several good sources on line. My favorite is at theUS National Institute of Standards and Technology, General Tables of Units of Measurement, though the sheer mass of information can be overwhelming.Its availablein PDFor via the Internet Archive in HTML.
Looking at such references, you may note that this article usescommon abbreviations like sec and hr , rather thanthe official abbreviations . That isdeliberate, since most students are more familiar with the longerforms. In scientific work, youd be expected to use the officialforms.
Unit Conversions With Multiple Conversion Factors
Sometimes multiple conversions are needed before we end up with the units we want. For example, How many minutes are there in 2 days? The process is the same, we just repeat the steps for every unit conversion we make.
Steps for Unit Conversions
The following video will show an example of using two conversion factors.
Time is one of the most commonly used conversions. Depending on what you are converting between, it is also a good example of sometimes needing more than one conversion factor.
Example:
How many minutes are there in two days?
Unless you know the conversion for minutes to days, this takes two conversion factors.
We can use the following equivalence statements to make our conversion factors.
1 day = 24 hours
1 hour = 60 minutes
First, we start with what we have which is two days.
Next, we use one of our equivalence statements to make the conversion factor that will allow us to cancel out days.
The conversion factor \ allows us to cancel days in the numerator with days in the denominator.
This leaves us with hours in the numerator. We still need another conversion factor to cancel out the hours. The conversion factor \ allows us to cancel out hours in the numerator with hours in the denominator.
Thus, two days = 2880 minutes.
Recommended Reading: Linear Algebra First Year University
Significant Figures In Conversions
How do conversion factors affect the determination of significant figures? Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact. Exact numbers are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact. In contrast, conversion factors that come from measurements or are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer.
Example \
Solution
We use the conversion factor that will cancel out the original unit, liters, and introduce the unit we are converting to, which is milliliters. The conversion factor that does this is the one on the right.
Exercise \
- Answer a
- Answer b
Example : Miles Per Hour To Kilometers Per Hour
This problem can be solved using either 1 mi =1.61 km or 1 km = 0.621 mi. Ill work it both ways,in parallel.
To start, write the original measurement as afraction:
11.6 mi ------- hr
Going from mi/hr to km/hr, you see that you end up with the samedenominator you started with, so only the numerator has to changeunits. In other words, this is just our old friend miles kilometers, with the per hour tagging along unchanged. So theconversion is the same one youve done before. Simply pick a fractionwith the desired units on top and the given units on the bottom:
11.6 mi 1.61 km 11.6 mi 1 km ------- × ------- or ------- × -------- hr 1 mi hr 0.621 mi
As you see, you can use either conversion factor, miles tokilometers or kilometers to miles. It doesnt matter because, byforming a fraction equal to 1, you automatically make the right choicebetween dividing and multiplying.
Going on to simplify the fractions, you have
11.6 × 1.61 mi km 11.6 mi km ----------------- or ----------- hr mi 0.621 hr mi
Either way, divide top and bottom by mi and you have
11.6 × 1.61 km 11.6 km -------------- or -------- hr 0.621 hr
Do the arithmetic to get 18.7 km/hr either way.
You May Like: Chapter 10 Test Form 2b Glencoe Geometry
Organic Chemistry Conversions Questionsexamplesproblems
Converting one organic compound to another organic compound by using one or more other organic compounds or reagents by a single or multiple steps is called an organic conversion.
Organic chemistry conversions of class 12 are major problems you have to do in examinations.
Organic conversions questions have important role in organic chemistry questions. If you practise these organic conversions properly, you can get full marks for these questions in your organic conversion class 12 examinations.
Conversion Factors From Different Units
Conversion factors can also be constructed for converting between different kinds of units. For example, density can be used to convert between the mass and the volume of a substance. Consider mercury, which is a liquid at room temperature and has a density of 13.6 g/mL. The density tells us that 13.6 g of mercury have a volume of 1 mL. We can write that relationship as follows:
13.6 g mercury = 1 mL mercury
This relationship can be used to construct two conversion factors:
Which one do we use? It depends, as usual, on the units we need to cancel and introduce. For example, suppose we want to know the mass of 16 mL of mercury. We would use the conversion factor that has milliliters on the bottom and grams on top so that our final answer has a unit of mass:
\ & \approx \mathrm \end \nonumber \]
In the last step, we limit our final answer to two significant figures because the volume quantity has only two significant figures the 1 in the volume unit is considered an exact number, so it does not affect the number of significant figures. The other conversion factor would be useful if we were given a mass and asked to find volume, as the following example illustrates.
Density can be used as a conversion factor between mass and volume.
Example \: Mercury Thermometer
A mercury thermometer for measuring a patients temperature contains 0.750 g of mercury. What is the volume of this mass of mercury?
Solution
\ & \approx 0.0551\ \rm \end \nonumber \]
Exercise \
Don’t Miss: What Do Double Brackets Mean In Math
List Of The Metric Or Si Base Units
The kilogram, meter, and second are the fundamental base units upon which the metric system is built, but seven units of measure are defined from which all the other units are derived:
- Kilogram: The kilogram is the base unit of mass.
- Meter or Metre: The meter is the unit of length or distance.
- Second: The second is the fundamental unit of time.
- Kelvin: The Kelvin is the metric unit of temperature.
- Mole: The mole is a unit for a quantity of a substance.
- Ampere: The Ampere is the unit of electric current.
- Candela: The candela is the unit of luminous intensity. The candela is sometimes called by its old name, the candle.
The names and symbols for the units are written with lowercase letters, except for Kelvin , which is capitalized because it was named in honor of Lord Kelvin, and Ampere , which is named for Andre-Marie Ampere.
The liter or litre is an SI derived unit of volume, equal to 1 cubic decimeter or 1000 cubic centimeters . The liter actually was a base unit in the original French metric system but is now defined in relation to length.
The spelling of liter and meter may be litre and metre, depending on your country of origin. Liter and meter are American spellings most of the rest of the world uses litre and metre.
Example : Computing Quantities From Measurement Results And Known Mathematical Relations
What is the density of common antifreeze in units of g/mL? A 4.00 qt sample of the antifreeze weighs 9.26 lb.
Since \large\text=\frac}} , we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A × unit conversion factor. The necessary conversion factors are given in Table 1.6: 1 lb = 453.59 g 1 L = 1.0567 qt 1 L = 1,000 mL. We can convert mass from pounds to grams in one step:
\large\text\cancel}\times \frac}}}=4.20\times ^\text
We need to use two steps to convert volume from quarts to milliliters.
Then, \large\text=\frac\times ^\text}^\text}=\text.
Alternatively, the calculation could be set up in a way that uses three unit conversion factors sequentially as follows:
\large\frac\cancel}}\cancel}}\times \frac}}}\times \frac\cancel}}}}\times \frac}}}=\text
Check Your Learning
What is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz ?
You May Like: What Does Vf Mean In Physics