Problem Solving, Unit Conversion, and Dimensional Analysis

The temperature of a substance changes by 1K. This is equal to a temperature change of

1∘C Looking at the image above, we can see that the freezing temperature of water on the Kelvin scale is 273.15 K, and its boiling temperature is 373.15 K. Notice that the numerical difference in these two reference temperatures is 100, the same is true for the Celsius scale. Water freezes at 0∘C and boils at 100∘C which is a difference of 100 in magnitude. This means that 1 K is equivalent to 1∘C in magnitude. This makes the conversion between the Celsius and Kelvin quite easy.

If we wish to convert a value in years into days, a conversion factor we could use would be:

365 days1 year The top and the bottom must be equal to each other, so 365 days and 1 year, as those represent the same length of time. In addition, days must be on top and years on the bottom so that we can appropriately cancel out the existing unit.

An object has a mass of 0.124kg and a volume of 1893mm3. What is its density in grams per cubic centimeter?

65.5 Divide the mass by the volume in order to find the density, and then use conversion factors to cancel the given units and leave the desired units. density=0.124kg1893mm3 To convert from kilograms to grams, use the relationship 1kg=1000g. The conversion factor 1000g1kg cancels kilograms and leaves grams. In the denominator, use the relationship 1cm=10mm. Since these are cubic units, use the conversion factor (10mm1cm)3=1000mm31cm3 to cancel cubic millimeters and leave cubic centimeters. Now calculate the density with the converted units. density=0.124kg1893mm3×1000g1kg×1000mm31cm3=65.505gcm3 When we multiply or divide, our answer should have the same number of significant figures as the number with the fewest significant figures. In this calculation, the numbers in the unit conversions are exact numbers. The mass of 0.124kg has the fewest number of significant figures, so the answer has three significant figures. Round the answer to 65.5gcm3.

Which of the following factors used in a dimensional analysis calculation have a finite number of significant figures that could determine the precision of the calculated value? Select the correct answer below:

A measurement of speed equaling (27.7 km/hour) Unit conversion factors are almost always exact numbers. For example, there are exactly 1000 m in 1 km because that is how the kilometer is defined. There is no uncertainty involved in the conversion factor, so it will not affect the number of significant figures in our answer. But when using actual measurements, the number of significant figures will have an impact on our calculation.

Britta traveled 1250 km on her road trip and used 209 L of gasoline, filling her tank at an average cost of 1.14 euros per liter. Pierce traveled 1405 km on his trip and used 175 L of gasoline, filling his tank at an average cost of 1.23 euros per liter. Who paid more per kilometer driven for their trip?

Britta First calculate Britta's cost per distance: 209 L×1.14€L×11250 km=0.191€km Next calculate Pierce's cost. 175 L×1.23€L×11405 km=0.153€km Therefore, Britta's cost per kilometer traveled was greater.

Choose the best explanation below for why dimensional analysis is also sometimes referred to as the factor label approach.

It is sometimes called the factor label approach because units are treated as a factor (along with a numeric factor) in a quantity like 25g, allowing us to subject the unit to algebraic operations as if they were variables. It is sometimes called the factor label approach because we can treat a quantity like 25g as a product of a numerical factor and a unit factor, allowing us to subject the unit to arithmetic operations as if they were variables. Dimensional analysis can be most helpful when helping to set up a problem involving multiplication or division, but it is not limited to multiplication and division: Adding or subtracting quantities, for example, requires they have the same unit, just like adding algebraic terms 5x and 6x, and this rule can be used to help set up or check a calculation.

Which of the following is the SI unit for temperature?

K The SI unit for temperature is Kelvin

Tyson Gay's best time to run 100.0 meters was 9.69 seconds. What was his average speed during this run, in miles per hour? (3.281ft=1 m)(1 mile=5280 ft)

To find the average speed, divide the distance traveled by the time and use conversion factors to cancel the given units, ms, and leave the desired units, mihr. speed=100.0m9.69s To convert from meters to miles, use the relationships 3.281ft=1m and 5280ft=1mi. The conversion factor 3.281ft1mcancels meters and leaves feet, and the conversion factor 1mi5280ft cancels feet and leaves miles. In the denominator, use the relationships 1min=60s and 1hr=60min. The conversion factor 60s1min cancels seconds and leaves minutes, and the conversion factor 60min1hr cancels minutes and leaves hours. Now multiply to find the answer. speed=100.0m9.69s×3.281ft1m×1mi5280ft×60s1min×60min1hr=23.08mihr When we multiply or divide, our answer should have the same number of significant figures as the number with the fewest significant figures, which in this case is the 9.69 seconds. Therefore, after rounding to three significant figures, we find that the speed in miles per hour is 23.1mihr.

The speed of light is around 6.706×108 miles per hour. What is the speed of light in units of miles per minute?

To find the speed of light in units of miles per minute, we must use the fact that 60min=1hr. 6.706×108mihr×1hr60min=1.118×107mimin Therefore, we find that the speed of light is approximately 1.118×107 miles per minute.

A thermometer reads 25∘C. If the temperature in the room is then increased by 16 K, what temperature will the thermometer read, in degrees Celsius?

We are told that initially, the temperature reads 25∘C, and that the temperature increases by 16K. To find the final temperature in degrees Celsius, we can first convert 25∘C to Kelvin. TK=T∘C+273.1525+273.15=298.15K Now, add the16Kchange to the temperature in Kelvin. 298.15K+16K=314.15K Lastly, convert the temperature back to degrees Celsius. T∘C=TK−273.15314.15K−273.15=41∘C It should be noted that this temperature can also be found by adding the temperature change, in Kelvin, to the temperature in Celsius. 25∘C+16K=41∘C This can be done because the magnitude of1Kis equal to the magnitude of1∘C. Notice that the freezing temperature of water in Kelvin is273.15K, and its boiling temperature373.15K. The numerical difference in these two reference temperatures is100, the same as for the Celsius scale.

When computing using dimensional analysis:

unit conversions can be done either simultaneously or separately We can perform unit conversions one at a time or many at a time, just be sure to keep in mind that this could create a discrepancy in the number of significant figures in your result, depending on when you decide to round your value.

Which of the following temperatures are equivalent to absolute zero (0 K)? Select all that apply.

−273.15∘C and −459.67∘F show absolute zero in the Celsius and Fahrenheit scales.