Converting One Metric Unit Of Energy Into A Different Metric Unit Of Energy
The most likely conversions you will have to make during your chemistry course will be between joules and kilojoules
multiply the number of kilojoules by 1000 to give an energy value in units of joules
energy = energy × 1000
divide the number of joules by 1000 to give an energy value in kilojoules
energy = energy ÷ 1000
Do you understand this?
But My Calculator Has A Convert Key
So? sooner or later youll have to convert a measurement withunits that arent in your calculator. At that point a lot of studentsstart to guess, and the more complex the units the more likely theyllguess wrong. If you understand what youre doing, using the techniqueson this page, youll have smooth sailing.
You can also use these same techniques to do currency conversions,which are probably not on your calculator because the rates fluctuate.See the practice problems.
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.
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Is This Really Multiplying By 1
But wait a minute! I hear you say. Youstarted with 4.5 and ended up with 270. How is that multiplying by 1?The answer is that we didnt start with a dimensionless pure number4.5, but with 4.5 hours and we didnt end up with a pure number 270but with 270 minutes. You should be able to convince yourself that ifyou bake a turkey for 270 minutes or 4½ hours, either way youwait the same length of time for dinner.
A number with units is different from a numberwithout units or with different units, just as 8x is differentfrom both 8 and 8y. Think of it this way: 3.27 dollars or eurosis the same as 327 cents, when you multiply by the carefully chosenform of 1, 100 cents/dollar or 100 cents/euro. If the top and bottomof the fraction are equal, the fraction equals 1 and the value aftermultiplying is the same as the value before multiplyingbutexpressed in different units, which of course is the whole point.
You might be asking yourself, Why all the fuss?Anybody knows that to convert hours to minutes you have to multiply by60. Well, yes, thats true. But I deliberately chose a simple exampleto show the method. Ill try to use more realistic examples from here on.
How To Pick A 1
You might be wondering how I knew to pick that particular fractionthat was equal to 1. There are just two simple steps:
Find a conversion factor between the given units and thedesired units, and write it as an equation.
Example: whether you have miles and need kilometers, oryou have kilometers and need miles, you can use eitherconversion factor between miles and kilometers, namely1 mi = 1.61 km or 1 km = 0.621 mi.Either equation will work equally well.
Convert that equation to a fraction with thedesired units on top and the given units on the bottom.More formally, divide both sides by the value of the side thatcontains the given units.
Example: To convert from miles to kilometers, you need afraction with the desired units on top and the givenunits on the bottom. Based on the above conversion factors, thatfraction must be either
1.61 km 1 km ------- or -------- 1 mi 0.621 mi
Those fractions look different, but remember that theyre bothequal to 1 and therefore they are just different forms of the samefraction. Either one will work just fine for the conversion.
Once youve selected a useful fractional form of 1, multiply theoriginal measurement by the fraction, and simplify.
Example: If the original measure was 15.7 miles, you wouldmultiply by either of the above fractions and obtain 25.3 km.
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Base And Derived Units
For most quantities, a unit is absolutely necessary to communicate values of that physical quantity. Imagine you need to buy some rope to tie something onto the roof of a car. How would you tell the salesperson how much rope you need without using some unit of measurement?
However, not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Therefore, only a small set of units is required. These units are called base units, and other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units.
Different systems of units are based on different choices of base units. The most widely used system of units is the International System of Units, or SI. There are seven SI base units, and all other SI units can be derived from these base units.
The seven base SI units are:
- Length: m
- Thermodynamic Temperature: K
- Amount of Substance: mol
- Luminous Intensity: cd
The base units of SI are actually not the smallest set possible smaller sets have been defined. For example, there are unit sets in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic fields are actually different manifestations of the same phenomenon.
- Area: m2
- Force: \text\cdot\text^2, or the Newton
- Energy: \text\cdot\text, or the Joule
Example : Miles Per Second To Miles Per Hour
Escape velocity from the earths surface is about 7.0 mi/sec. What isthat in mi/hr?
Here again, youre converting only one unit, seconds to hours , and the miles per is just along for the ride. But whats different here is that the units youre converting are in the denominator of the fraction, not in the numerator. Look what happens if you apply the old rule of desired units on the top: | 7.0 mi 1 hr------ × -------- sec 3600 sec |
and you end up with | 7.0 mi hr------------3600 sec sec |
This is no good: you cant simplify this fraction to remove theseconds . Rule 2 in picking a fraction, asoriginally stated, only applied when the units to be converted were inthe top of the fraction .
Heres the more general form of Rule 2,the form that will always work:when picking your fraction that equals 1,put the given units in the opposite position from where they are in the originalmeasurement. If the original measurement had the given units on thetop, your 1-fraction will have them on the bottom but if the givenunits are on the bottom of the original measurement then your1-fraction must have them in the top. Do this so that you can dividetop and bottom by the given units when simplifying.
Lets come back to the example, 7.0 mi/secconverted to mi/hr, and do it the right way:
Write the conversion equation: |
about 25,000 mi/hr |
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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 \]
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:
Concept Map
Calculation
\ & = 5.47 \times 10^7\ \rm \end \]
In each step, the previous unit is canceled and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left.
Either methodâone step at a time or all the steps togetherâis acceptable. If you do all the steps together, the restriction for the proper number of significant figures should be done after the last step. As long as the math is performed correctly, you should get the same answer no matter which method you use.
Example \
Convert 43.007 mg to kilograms in one multistep calculation.
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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\frac}}\text=\text\frac}}
Several other commonly used conversion factors are given in Table 1.
Table 1. Common Conversion Factors | |
---|---|
Length | |
1 in. = 2.54 cm | 1 qt = 0.94635 L |
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\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.
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Recap Of The Procedure
In a nutshell, do all conversions of units by multiplying theoriginal measurement by a well-chosen form of the number 1. A bit lessbriefly:
Find the conversion factor for thegiven and desired units, and write it asa fraction with the given units in the opposite position from the originalmeasurement. The value of that fraction is 1, sincethe top and bottom are equal.
If the given units are raised to a power,raise the conversion fraction to that same power.
Multiply the original measurementby the conversion fraction, and simplify.
Calculate The Height Of The Person 53 Tall In Meters
The conversion of height from feet to meters is a two-step process. First, convert the number of feet to meters, and then convert the number of inches to meters.Converting feet to meters, we get5 ft × 0.305 = 1.53 metersNow, converting the inches to centimeters, we get3 inches × 2.54 = 7.62 cm = 0.0762 metersAdding these two together, we get1.53 meters + 0.0762 meters = 1.6062 meters.Therefore, the height in meters is 1.6062 meters.
For further reading:
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Example : Kilometers/hour To Meters Per Second
A race car has a top speed of 310 km/hr. What is that in m/sec?For this example youll combinechaining multiple conversionswith the new and more general form of Rule 2for picking the fraction.
You have two conversions to do, kilometers to meters and hours to seconds. You know the conversion factors: | 1 hr = 3600 sec1 km = 1000 m |
In converting km to m, the given units are on top, so in the conversion fraction km will be on the bottom. By contrast, in converting hr to sec, the given units are on the bottom so the conversion fraction will have hr on the top. To do two conversions, you multiply by two fractions : | |
310 km/hr = 86 m/sec |
Key Conversion Factors To Know
Normally, these conversion factors are taught as a part of their math program. But, I also recommend that have your students memorize several basic conversion factors between the two systems as part of their science plan.
Heres what every student should know:
- Pounds to Kilograms: 1 kg = 2.2 lb
- Gallons to Liters: 1 gal = 3.785 L
- Feet to Meters: 1 foot = 0.305 m
- Miles to Kilometers: 1 mi = 1.61 km
- Cups to Milliliters: 1 c = 240 mL
- Inches to Centimeters: 1 in = 2.54 cm
- Ounces to Grams: 1 oz = 28.3 g
With the global flow of information that occurs these days, it is very important for students to learn these most basic conversion factors.
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Worked Examples Of Converting Calories To Joules And Joules To Calories
Question 1: Convert 100 calories to joules
Solution:
multiply both sides of the equation by 100:
100 × 1 calorie = 100 × 4.18 J
100 cal = 418 J
Question 2: Convert 12.0 kilocalories to kilojoules
Solution:
multiply both sides of the equation by 12.0
12.0 × 1 kilocalorie = 12.0 × 4.18 kJ
12.0 kcal = 50.2 kJ
Question 3: Convert 150 kilocalories to joules
Solution:
Calculate the number of kilojoules in 150 kilocalories:
1 kilocalorie = 4.18 kilojoules
multiply both sides of the equation by 150
150 × 1 kilocalorie = 150 × 4.18 kJ
150 kcal = 627 kJ
Calculate the number of joules in 622 kJ
1 kilojoule = 1000 joules
multiply both sides of the equation by 622
627 × 1 kJ = 627 × 1000 J
627 kJ = 627 000 J
Write the answer to the question:
150 kcal = 627,000 J
Question 4: Convert 10 joules to calories
Solution:
multiply both sides of the equation by 10
10 × 1 J = 10 × 0.239 cal
10 J = 2.39 cal
Question 5: Convert 1.2 kilojoules to kilocalories
Solution:
multiply both sides of the equation by 1.2
1.2 × 1 kJ = 1.2 × 0.239 kcal
1.2 kJ = 0.287 kcal
Question 6: Convert 1500 joules to kilocalories
Solution:
Calculate the number of calories in 1500 joules
1 joule = 0.239 calories
multiply both sides of the equation by 1500
1500 × 1 J = 1500 × 0.239 cal
1500 J = 359 calories
Calculate the number of kilocalories in 539 calories
1000 calories = 1 kilocalorie
Divide both sides of the equation by 1000
1000
multiply both sides of the equation by 539
359 × 1 calorie = 359 × 0.001 kilocalories
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
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Unit Conversion And Conversion Factors
A unit conversion expresses the same property as a different unit of measurement. For instance, time can be expressed in minutes instead of hours, while distance can be converted from miles to kilometers, or feet, or any other measure of length. Often measurements are given in one set of units, such as feet, but are needed in different units, such as chains. A conversion factor is a numeric expression that enables feet to be changed to chains as an equal exchange.
Javier constructed 41 chains of dozer line.Notice that Table 2.2 has two conversions for each set of units. When setting up the cancellation table, it is not important which conversion factor is used. What is important is that the appropriate units cancel so that the correct end result is achieved.
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:
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