Chemisty Unit 9 Worksheet 1 Fases Again Chemistry Unit 9 Worksheet 1 Gases Again Answers

Converting Units

Learning Objective

Convert from one unit of measurement to some other unit of the aforementioned type.

In section "Expressing Units", we showed some examples of how to replace initial units with other units of the same type to get a numerical value that is easier to embrace. In this section, nosotros will formalize the process.

Consider a simple instance: how many feet are at that place in 4 yards? Most people will almost automatically answer that at that place are 12 feet in 4 yards. How did you make this determination? Well, if there are 3 feet in ane chiliad and there are iv yards, so there are iv × 3 = 12 anxiety in 4 yards.

This is right, of course, merely it is informal. Let us formalize it in a way that can be applied more generally. Nosotros know that 1 yard (yd) equals 3 anxiety (ft):

1 yd = 3 ft

In math, this expression is called an equality. The rules of algebra say that y'all tin change (i.east., multiply or divide or add or subtract) the equality (as long as you don't split past nothing) and the new expression will still be an equality. For example, if nosotros divide both sides by 2, we go

$$\frac{\mathrm{ane}}{2}\text{ yd}=\frac{3}{2}\text{ ft}$$

We meet that half of a yard equals 3/two, or ane and a one-half, feet—something we also know to exist true, so the in a higher place equation is notwithstanding an equality. Going back to the original equality, suppose we divide both sides of the equation by 1 chiliad (number and unit):

$$\frac{\text{one yd}}{\text{1 yd}}=\frac{\text{3 ft}}{\text{ane yd}}$$

The expression is still an equality, by the rules of algebra. The left fraction equals 1. It has the same quantity in the numerator and the denominator, so it must equal ane. The quantities in the numerator and denominator cancel, both the number and the unit:

$$\frac{\cancel{\text{i}}\cancel{\text{ yd}}}{\cancel{\text{one}}\cancel{\text{ yd}}}=\frac{\text{3 ft}}{\text{i yd}}$$

When everything cancels in a fraction, the fraction reduces to one:

$$\mathrm{one}=\frac{\text{three ft}}{\text{one yd}}$$

We have an expression, $\frac{\text{3 ft}}{\text{ane yd}}$ , that equals 1. This is a strange fashion to write 1, but it makes sense: 3 ft equal 1 yd, so the quantities in the numerator and denominator are the aforementioned quantity, but expressed with dissimilar units. The expression $\frac{\text{3 ft}}{\text{i yd}}$ is called a conversion factor , and it is used to formally alter the unit of measurement of a quantity into some other unit. (The procedure of converting units in such a formal fashion is sometimes called dimensional analysis or the gene label method.)

To come across how this happens, let us start with the original quantity:

4 yd

Now let united states of america multiply this quantity past 1. When you multiply anything by i, y'all don't alter the value of the quantity. Rather than multiplying past only 1, let united states of america write 1 every bit $\frac{\text{3 ft}}{\text{one yd}}$ :

$$\text{4 yd}\times \frac{\text{3 ft}}{\text{i yd}}$$

The four yd term tin be thought of equally $\frac{\text{4 yd}}{1}$ ; that is, it can be thought of as a fraction with 1 in the denominator. Nosotros are essentially multiplying fractions. If the same affair appears in the numerator and denominator of a fraction, they cancel. In this case, what cancels is the unit one thousand:

$$\text{4}\cancel{\text{ yd}}\times \frac{\text{3 ft}}{\text{1}\cancel{\text{ yd}}}$$

That is all that we tin abolish. Now, multiply and divide all the numbers to get the final respond:

$$\frac{4\times \mathrm{three}\text{ ft}}{1}=\frac{\text{12 ft}}{\mathrm{i}}=12\text{ ft}$$

Again, nosotros get an answer of 12 ft, just as we did originally. But in this example, we used a more formal procedure that is applicable to a multifariousness of problems.

How many millimeters are in xiv.66 yard? To respond this, we demand to construct a conversion gene between millimeters and meters and apply it correctly to the original quantity. We commencement with the definition of a millimeter, which is

1 mm = 1/one,000 m

The ane/1,000 is what the prefix milli- means. Most people are more comfy working without fractions, so nosotros will rewrite this equation by bringing the 1,000 into the numerator of the other side of the equation:

1,000 mm = 1 one thousand

Now nosotros construct a conversion factor past dividing one quantity into both sides. But at present a question arises: which quantity do we divide by? It turns out that we have ii choices, and the two choices volition give us different conversion factors, both of which equal 1:

$$\frac{\mathrm{i,000}\text{ mm}}{\mathrm{ane,000}\text{ mm}}=\frac{\mathrm{one}\text{ m}}{\mathrm{1,000}\text{ mm}}\text{or}\frac{\mathrm{1,000}\text{ mm}}{1\text{ yard}}=\frac{1\text{ m}}{\mathrm{i}\text{ k}}$$

$$1=\frac{1\text{ m}}{\mathrm{1,000}\text{ mm}}\text{or}\frac{\mathrm{1,000}\text{ mm}}{1\text{ yard}}=$$ $$1$$ 

Which conversion factor do nosotros use? The reply is based on what unit you want to go rid of in your initial quantity. The original unit of our quantity is meters, which we want to convert to millimeters. Because the original unit is assumed to exist in the numerator, to get rid of it, we want the meter unit in the denominator; so they will cancel. Therefore, we will utilise the second conversion factor. Canceling units and performing the mathematics, we go

$$\text{14}.\text{66}\cancel{\text{ thou}}\times \frac{\mathrm{1,000}\text{ mm}}{\mathrm{i}\cancel{\text{ m}}}=\mathrm{14,660}\text{ mm}$$

Note how chiliad cancels, leaving mm, which is the unit of involvement.

The power to construct and apply proper conversion factors is a very powerful mathematical technique in chemical science. You need to principal this technique if you are going to exist successful in this and future courses.

Example 7

Catechumen 35.9 kL to liters. Convert 555 nm to meters. Solution 1. We will utilize the fact that 1 kL = 1,000 Fifty. Of the 2 conversion factors that can exist defined, the i that will piece of work is 1,000 L. Applying this conversion gene, we go $$\text{<span>35.nine </span>}\cancel{\text{<span> kL</span>}}\text{<span></span>}<span>\times </span>\text{<span></span>}\frac{\mathrm{<span>1,000</span>}\text{<span> Fifty</span>}}{\mathrm{<span>1</span>}\cancel{\text{<span> kL</span>}}}<span>=</span>\mathrm{<span>\; 35.900\; 50\; </span>}$$ 2. We will use the fact that 1 nm = one/i,000,000,000 m, which we volition rewrite as 1,000,000,000 nm = one chiliad, or 109 nm = 1 m. Of the ii possible conversion factors, the advisable one has the nm unit of measurement in the denominator: 1 m. Applying this conversion factor, nosotros get $$\text{<span>555 </span>}\cancel{\text{<span> nm</span>}}\text{<span></span>}<span>\times </span>\text{<span></span>}\frac{\mathrm{<span>1\; </span>}\text{<span>m</span>}}{\mathrm{<span>10⁹</span>}\cancel{\text{<span> nm</span>}}}<span>=</span>\mathrm{<span>0.000000555\; </span>}\text{<span>thou = 5.55 x 10¯⁷m</span>}$$ In the terminal step, we expressed the answer in scientific notation. Test Yourself Convert 67.08 μL to liters. Convert 56.8 k to kilometers. Answers 6.708 × 10−5 L v.68 × 10−two km

What if we have a derived unit that is the product of more than one unit, such as m²? Suppose we desire to catechumen foursquare meters to square centimeters? The key is to remember that grand² means m × k, which means nosotros have 2 meter units in our derived unit of measurement. That means we take to include two conversion factors, one for each unit. For instance, to catechumen 17.6 m^two to foursquare centimeters, we perform the conversion as follows:

$$17.6{\text{ m}}^2=17.6\left(\cancel{\text{ yard}}\times \cancel{\text{ m}}\right)\times \frac{100\text{ cm}}{\mathrm{i}\cancel{\text{ m}}}\times \frac{100\text{ cm}}{1\cancel{\text{ k}}}=\mathrm{176,000}\text{ cm}\times \text{cm}=\mathrm{one.76}\times {10}^5{\text{ cm}}^{\mathrm{two}}$$

Example 8

How many cubic centimeters are in 0.883 mthree? Solution With an exponent of 3, we have three length units, so by extension we need to apply three conversion factors between meters and centimeters. Thus, nosotros have 0.883 thousand 3 × 100 cm 1 k × 100 cm 1 m × 100 cm 1 thou = 883,000 cm three = 8.83 × 10 5 cm 3 You lot should demonstrate to yourself that the three meter units do indeed cancel. Exam Yourself How many cubic millimeters are nowadays in 0.0923 one thousand3? Answer ix.23 × 107 mm3

Suppose the unit of measurement you desire to convert is in the denominator of a derived unit; what then? And then, in the conversion factor, the unit you desire to remove must be in the numerator. This will cancel with the original unit of measurement in the denominator and introduce a new unit in the denominator. The following instance illustrates this situation.

Instance 9

Convert 88.four g/min to meters/second. Solution Nosotros desire to modify the unit in the denominator from minutes to seconds. Because in that location are 60 seconds in one minute (lx s = one min), we construct a conversion factor then that the unit nosotros want to remove, minutes, is in the numerator:  Apply and perform the math: Detect how the 88.4 automatically goes in the numerator. That's considering whatever number can be thought of as beingness in the numerator of a fraction divided by one. Examination Yourself Convert 0.203 chiliad/min to meters/2d. Answer 0.00338 k/s or 3.38 × ten−three m/south

Sometimes there will be a demand to catechumen from one unit with one numerical prefix to another unit with a unlike numerical prefix. How do we handle those conversions? Well, you could memorize the conversion factors that interrelate all numerical prefixes. Or y'all can go the easier road: first convert the quantity to the base unit, the unit of measurement with no numerical prefix, using the definition of the original prefix. Then convert the quantity in the base unit to the desired unit using the definition of the second prefix. You tin can exercise the conversion in two separate steps or as one long algebraic step. For example, to convert two.77 kg to milligrams:

Alternatively, information technology can be done in a single multistep procedure:

You become the same answer either fashion.

Example x

How many nanoseconds are in 368.09 μs? Solution You can either practice this as a ane-step conversion from microseconds to nanoseconds or catechumen to the base of operations unit outset and then to the final desired unit. We will use the second method here, showing the two steps in a single line. Using the definitions of the prefixes micro- and nano-, $$368.09\cancel{\mu \text{s}}\times \frac{\text{1}\cancel{\text{ due south}}}{\mathrm{1,000,000}\cancel{\mu \text{s}}}\times \frac{\mathrm{ane,000,000,000}\text{ ns}}{1\cancel{\text{ southward}}}=\mathrm{368,090}\text{ ns}=\text{3}\text{.6809}\times {\text{ten}}^5\text{ ns}$$ Test Yourself How many milliliters are in 607.8 kL? Reply half dozen.078 × 10eight mL

When because the meaning figures of a final numerical reply in a conversion, at that place is one of import case where a number does not bear upon the number of significant figures in a final answer—the so-called exact number. An exact number is a number from a defined relationship, not a measured 1. For example, the prefix kilo- means ane,000—exactly 1,000, no more or no less. Thus, in constructing the conversion factor

$$\frac{\mathrm{1,000}\text{ g}}{1\text{ kg}}$$

neither the 1,000 nor the ane enter into our consideration of significant figures. The numbers in the numerator and denominator are defined exactly past what the prefix kilo- means. Another way of thinking about it is that these numbers can exist thought of equally having an infinite number of meaning figures, such equally

$$\frac{\mathrm{1,000.0000000000}\dots \text{ g}}{1.0000000000\dots \text{ kg}}$$

The other numbers in the calculation will determine the number of significant figures in the final reply.

Instance eleven

A rectangular plot in a garden has the dimensions 36.7 cm by 128.8 cm. What is the area of the garden plot in square meters? Limited your answer in the proper number of significant figures. Solution Surface area is divers as the product of the two dimensions, which we so accept to convert to square meters and express our concluding answer to the correct number of pregnant figures, which in this example will exist 3 $36.7\cancel{c\mathrm{one\; thousand}}\times 128.8\cancel{cm}\times \frac{1m}{100\cancel{cm}}\times \frac{\mathrm{one}k}{100\cancel{cm}}=0.472696m^{\mathrm{two}}=0.473{\mathrm{1000}}^{\mathrm{two}}$ The ane and 100 in the conversion factors do not touch the determination of significant figures because they are exact numbers, defined by the centi- prefix. Test Yourself What is the volume of a block in cubic meters whose dimensions are ii.one cm × 34.0 cm × 118 cm? Answer 0.0084 grand3

Chemistry Is Everywhere: The Gimli Glider

On July 23, 1983, an Air Canada Boeing 767 jet had to glide to an emergency landing at Gimli Industrial Park Airport in Gimli, Manitoba, considering it unexpectedly ran out of fuel during flying. There was no loss of life in the course of the emergency landing, only some minor injuries associated in role with the evacuation of the craft later landing. For the remainder of its operational life (the plane was retired in 2008), the shipping was nicknamed "the Gimli Glider."

The 767 took off from Montreal on its way to Ottawa, ultimately heading for Edmonton, Canada. About halfway through the flight, all the engines on the plane began to close down because of a lack of fuel. When the final engine cut off, all electricity (which was generated by the engines) was lost; the plane became, essentially, a powerless glider. Captain Robert Pearson was an experienced glider pilot, although he had never flown a glider the size of a 767. Kickoff Officer Maurice Quintal quickly determined that the aircraft would not be able get in to Winnipeg, the adjacent big airdrome. He suggested his old Royal Air Strength base at Gimli Station, one of whose runways was still beingness used as a customs airport. Between the efforts of the pilots and the flying coiffure, they managed to get the airplane safely on the ground (although with buckled landing gear) and all passengers off safely.

What happened? At the time, Canada was transitioning from the older English system to the metric organisation. The Boeing 767s were the first aircraft whose gauges were calibrated in the metric system of units (liters and kilograms) rather than the English system of units (gallons and pounds). Thus, when the fuel approximate read 22,300, the approximate meant kilograms, simply the footing crew mistakenly fueled the plane with 22,300 pounds of fuel. This ended upward being just less than half of the fuel needed to make the trip, causing the engines to quit most halfway to Ottawa. Quick thinking and extraordinary skill saved the lives of 61 passengers and viii crew members—an incident that would not take occurred if people were watching their units.

Key Takeaways

Units tin can exist converted to other units using the proper conversion factors. Conversion factors are synthetic from equalities that relate two different units. Conversions can be a single step or multistep. Unit conversion is a powerful mathematical technique in chemical science that must be mastered. Exact numbers exercise not impact the decision of significant figures.

Exercises

Write the two conversion factors that be betwixt the 2 given units. milliliters and liters microseconds and seconds kilometers and meters Write the 2 conversion factors that be between the 2 given units. kilograms and grams milliseconds and seconds centimeters and meters Perform the following conversions. 5.4 km to meters 0.665 m to millimeters 0.665 g to kilometers Perform the following conversions. 90.half-dozen mL to liters 0.00066 ML to liters 750 L to kiloliters Perform the following conversions. 17.8 μg to grams 7.22 × ten2 kg to grams 0.00118 thou to nanograms Perform the following conversions. 833 ns to seconds 5.809 southward to milliseconds 2.77 × 10half dozen s to megaseconds Perform the following conversions. ix.44 m2 to square centimeters 3.44 × 10eight mm3 to cubic meters Perform the following conversions. 0.00444 cm3 to cubic meters 8.eleven × 10two mtwo to square nanometers Why would information technology be inappropriate to catechumen foursquare centimeters to cubic meters? Why would it exist inappropriate to convert from cubic meters to cubic seconds? Perform the post-obit conversions. 45.0 m/min to meters/second 0.000444 m/s to micrometers/second 60.0 km/h to kilometers/second Perform the following conversions. 3.4 × 102 cm/s to centimeters/infinitesimal 26.6 mm/southward to millimeters/60 minutes xiii.7 kg/L to kilograms/milliliters Perform the following conversions. 0.674 kL to milliliters 2.81 × 1012 mm to kilometers 94.5 kg to milligrams Perform the following conversions. 6.79 × 10−6 kg to micrograms i.22 mL to kiloliters 9.508 × 10−ix ks to milliseconds Perform the following conversions. 6.77 × x14 ms to kiloseconds 34,550,000 cm to kilometers Perform the following conversions. 4.701 × 1015 mL to kiloliters 8.022 × 10−11 ks to microseconds Perform the post-obit conversions. Note that you will accept to convert units in both the numerator and the denominator. 88 ft/south to miles/hour (Hint: use 5,280 ft = 1 mi.) 0.00667 km/h to meters/2nd Perform the following conversions. Note that you will accept to convert units in both the numerator and the denominator. iii.88 × xtwo mm/s to kilometers/hour one.004 kg/L to grams/milliliter What is the area in square millimeters of a rectangle whose sides are 2.44 cm × 6.077 cm? Express the answer to the proper number of meaning figures. What is the volume in cubic centimeters of a cube with sides of 0.774 m? Limited the answer to the proper number of significant figures. The formula for the area of a triangle is 1/two × base of operations × height. What is the expanse of a triangle in square centimeters if its base is ane.007 yard and its peak is 0.665 chiliad? Express the answer to the proper number of significant figures. The formula for the expanse of a triangle is i/ii × base × meridian. What is the expanse of a triangle in foursquare meters if its base is 166 mm and its meridian is 930.0 mm? Express the respond to the proper number of significant figures.

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Source: https://guides.hostos.cuny.edu/che105

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