 # Unit 1 - Temps, SFs, Dimensional Analysis

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Unit 1 - Temps, SFs, Dimensional Analysis
4/15/2017 Temperature Scales and Conversions Boiling point of water 100°C K 212°F freezing point of water K 0°C 32°F K - 40°C - 40°F Absolute zero 0 K °C °F Kelvin Celsius T(°C) = T(K) – T(°C) = 𝟓 𝟗 (T(°F) – 32) Fahrenheit T(°F) = 𝟗 𝟓 T(°C) + 32

Converting T(°F) to T(K)
Convert body temperature 98.6°F to kelvin. T(°𝑭) = 𝟗 𝟓 𝑻(°𝑪) + 𝟑𝟐 𝟗𝟖.𝟔= 𝟗 𝟓 𝑻(°𝑪)+ 𝟑𝟐 𝟗𝟖.𝟔−𝟑𝟐 = 𝟗 𝟓 𝑻(°𝑪) 𝟔𝟔.𝟔 𝟓 𝟗 = 𝑻(°𝑪) 98.6°F = 37°C 𝐓(𝐊) =𝑻(°𝑪)+𝟐𝟕𝟑.𝟏𝟓 𝑻(𝑲)=𝟑𝟕+𝟐𝟕𝟑.𝟏𝟓 98.6°F = 310. K, or T = 310. K

Mass and volume are extensive, physical properties.
mass volume Density = Mass and volume are extensive, physical properties. Density is an intensive physical property, the value of which changes with temperature. SI units of density are kg/m3. Commonly used units of density are g/cm3. The density of water at 25°C is 1.00 g/cm3. Materials with densities greater than water will sink in water (e.g. Au, density g/cm3). Materials with densities less than water will float on water (balsa wood, density 0.16 g/cm3).

1 mL = 1 cm3 mass volume Density = density = 26.113 g
I measure the mass of a chunk of metal and find it to be g. I measure its volume by displacement and find it to be 10.0 mL. What is the density of the metal? density = g mL = g cm3 = 2.61 g/cm3 I measure the mass of a cube of metal and find it to be g. I measure one of its sides and find it to be 0.52 cm. What is the density of the metal? volume of a cube = (0.52 cm)3 = cm3 1 mL = 1 cm3 density = g cm3 = 1.9 x 102 g/cm3

mass volume Density = Density is used to convert mass to volume. How many milliliters will 15.0 g of ethanol occupy when its density is 0.79 g/cm3? 15.0 g x conversion factor = volume in mL Remember: 0.79 g = g cm mL 15.0 g x mL = 19 mL 0.79 g Density is used to convert volume to mass. What is the mass of cm3 of ethanol? 100.0 cm3 x conversion factor = mass in g 100.0 cm3 x 0.79 g = 79 g cm3

Density The red liquid is water with food color. What can be said about the density of the yellow liquid?

The Scientific Method and Measured Data
Central to the scientific method is the accumulation and study of measured data. A measurement consists of two parts: a number and a unit. Both must be reported correctly. A systematic set of units (SI units and the metric prefixes) allows scientists from different areas to communicate easily. A systematic way to report the number measured (by using the correct number of significant figures) communicates how good you think your measurements are.

Types of Numbers (Data)
Exact numbers (data) obtained from counting some conversions (e.g cm = 1 inch, exactly) Inexact most measured data Significant figures apply to inexact numbers!

Uncertainty in Measured Data
Measured data is written to convey two (2) things! the magnitude of the measurement the extent of its reliability Worker #1 reports a mass of 12 g Worker #2 reports a mass of g 12 g means 12 ± 1 g g means ± g 12 g has 2 significant figures. g has 6 significant figures. g is the more certain (reliable) number. The more significant figures a measurement has, the more certain it is.

Measured Values: Accuracy vs. Precision
accurate and precise precise but not accurate not accurate not precise Accuracy is how close your measured value is to the right value (can be shown by % error). Precision is how well you can reproduce your measurement (can be shown by standard deviation).

Recording Data to the Correct Number of Significant Figures
The number of SFs in a measured value is equal to the number of known digits plus one uncertain digit. 22°C 22°C 21°C 21°C recorded value = 21.6°C recorded value = 21.68°C

Making Measurements in the Lab: Recording Volumetric Data to the Correct Number of Significant Figures - Glassware with Graduations Example B Example A 1. If the glassware is marked every 10 mLs, the volume you record should be in mLs. (Example A) 2. If the glassware is marked every 1 mL, the volume you record should be in tenths of mLs. 3. If the glassware is marked every 0.1 mL, the volume you record should be in hundredths of mLs. (Example B) 0 mL 30 mL 20 mL 1 mL 10 mL 30-mL beaker: the volume you write in your lab report should be 13 mL 2 mL Buret marked in 0.1 mL: you record volume as 0.67 mL

Unit 1 - Temps, SFs, Dimensional Analysis
4/15/2017 Making Measurements in the Lab: Recording Volumetric Data to the Correct Number of Significant Figures - Volumetric Glassware Look on the glassware for written indication of the precision of the volumetric flask or pipet. On this volumetric flask is written 500mL ± 0.2 mL. You would record the volume of the liquid in this flask as mL.

Trailing zeros MUST be recorded.
Making Measurements in the Lab: Recording Masses to the Correct Number of Significant Figures This one is easy: record EVERY number (especially zeros) that appears on the display of the electronic balance. Trailing zeros MUST be recorded.

How to Count Significant Figures
All nonzero digits are significant (1.23 has 3 SFs). All zeros between nonzero digits are significant (1.003 has 4 SFs). Leading zeros are NEVER significant (0.01 has 1 SF). Trailing zeros when a decimal point is present are significant ( has 3 SFs and 180. has 3 SFs.) Trailing zeros when no decimal point is shown are not significant. (180 has 2 SFs.)

Scientific Notation An unambiguous way to show the number of significant figures (SFs) in your data Numbers are written as the product of a number greater than or equal to 1 and less than 10 and a power of 10. Measurement in scientific notation #SFs mi/s m 512.1 x 101 g x 105 mi/s x 10-3 m 5.121 x 103 g 6 5 4

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data
Multiplication/Division: The answer contains the same number of SFs as the measurement with the fewest SFs. 25.2 x 6.1 = (but only 2 SFs are allowed) = 1.5 x 102 (correct answer) 25.2 = (on my calculator) = 7.31 (correct answer) 25.2 x 6.1 = (on my calculator) = 45 (correct answer)

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data
Addition/Subtraction: The answer contains the same number of digits to the right of the decimal as that of the measurement with the fewest number of decimal places. + 25.2 28.3 (correct answer) 3 SFs 0.10 (correct answer) 2 SFs Calculators do NOT know these rules. It’s up to you to apply them!

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data
Addition/Subtraction: Dealing with numbers with no decimal places. Convert both numbers to exponential notation with the same power of ten, and then use the decimal place rule. 286.4 x x 103 = ? x 105 x 105 x 105 286.3 x 105 (correct answer) 4 SFs

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data
Addition/Subtraction: Dealing with numbers with no decimal places. Write out the numbers and underline uncertain digit. 286.4 x x 103 = ? 28,640,000 (uncertain in 10,000 place) ,100 (uncertain in 100 place) 28,631,900 (take the uncertain digit farthest to the left) 28,630,000 or x 107 4 SFs

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data
Combined operations: Do the add/subtract first, carrying all digits, then do the multiply/divide. The only time you round is at the very end of the calculation. % difference = 100 x (your value - accepted value) accepted value an exact number…it does not affect SFs

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data
Find the percent difference between and an accepted value of % difference = 100 x ( ) 3.025 First, subtract from 3.015: result has 2 SFs Second, multiply by 100 and divide by (4 SFs) (in either order): Finally, round to 2 SFs: %

Converting Measurements Using Dimensional Analysis
Helps ensure that the answers to problems have the proper units. Uses conversion factors to reach the proper units. A conversion factor is a fraction whose numerator and denominator are the same quantity expressed in different units. 2.54 cm = 1 inch (2.54 cm and 1 inch are the same length.) 2.54 cm 1 inch and are conversion factors. 1 inch cm

Converting Measurements Using Dimensional Analysis
Convert 16.7 gallons to liters. 3.785 L 16.7 gal x = 63.2 L 1 gal Convert liters to cubic meters (m3). 𝟎.𝟖𝟒𝟕𝟒 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏 𝑳 𝟏 𝒄𝒎 𝟑 𝟏𝒎𝑳 𝟏𝟎 −𝟐 𝒎 𝟏𝒄𝒎 𝟏𝟎 −𝟐 𝒎 𝟏𝒄𝒎 𝟏𝟎 −𝟐 𝒎 𝟏𝒄𝒎 =𝟖.𝟒𝟕𝟒𝒙 𝟏𝟎 −𝟒 𝒎 𝟑 OR 𝟎.𝟖𝟒𝟕𝟒 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏 𝑳 𝟏 𝒄𝒎 𝟑 𝟏𝒎𝑳 𝟏𝟎 −𝟐 𝒎 𝟏𝒄𝒎 𝟑 =𝟖.𝟒𝟕𝟒𝒙 𝟏𝟎 −𝟒 𝒎 𝟑

Density Problem - Where to start?
The density of air at ordinary atmospheric pressure and 25°C is 1.19 g/L. What is the mass, in pounds, of the air in a room that measures 12.5ft x 15.5ft x 8.0ft? The volume of the room is (V = l w h): 12.5 ft x 15.5 ft x 8.0 ft = 1550 ft3 (We treat sig figs at the end.) The temptation is to start with the density. But density has units in the numerator AND denominator: 𝒈 (𝒖𝒏𝒊𝒕 𝒐𝒇 𝒎𝒂𝒔𝒔) 𝑳 (𝒖𝒏𝒊𝒕 𝒐𝒇 𝒗𝒐𝒍𝒖𝒎𝒆) Our answer should only contain a unit of mass (lbs).

Density Problem - pay attention to units!
The density of air at ordinary atmospheric pressure and 25°C is 1.19 g/L. What is the mass, in pounds, of the air in a room that measures 12.5ft x 15.5ft x 8.0ft? Start with the volume (it has units only in the numerator). 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝒎𝒂𝒔𝒔 𝒖𝒏𝒊𝒕𝒔 𝒗𝒐𝒍𝒖𝒎𝒆 𝒖𝒏𝒊𝒕𝒔 =?𝒍𝒃𝒔 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝟏.𝟏𝟗 𝒈 𝑳 =?𝒍𝒃𝒔 Clearly, we need to change g to lbs and L to ft3.

Density Problem - Using conversion factors
The density 1.19 g/L is in metric units (a close cousin to SI units), but lbs (pounds) and cubic feet are English units. For length, a good conversion factor is 1 in = 2.54 cm. Since volume comes from the cube of a length, we have to cube the conversion factor: 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝟏.𝟏𝟗 𝒈 𝑳 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏 𝒎𝑳 𝒄𝒎 𝟑 (𝟐.𝟓𝟒 𝒄𝒎 ) 𝟑 (𝒊𝒏 ) 𝟑 (𝟏𝟐 𝒊𝒏 ) 𝟑 (𝒇𝒕 ) 𝟑 =?𝒍𝒃𝒔 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝟏.𝟏𝟗 𝒈 𝑳 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏 𝒎𝑳 𝒄𝒎 𝟑 𝟐.𝟓𝟒 𝟑 𝒄𝒎 𝟑 𝒊𝒏 𝟑 𝟏𝟐 𝟑 𝒊𝒏 𝟑 𝒇𝒕 𝟑 =?𝒍𝒃𝒔

Density Problem - Dimension Analysis Form
If we stopped here, we would have a mass. But the units would be grams, not lbs. One final conversion is needed: 1 lb = g (For a conversion that looks like this, the “1” is treated as exact, so the number of sig figs here would be 4.) 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝟏.𝟏𝟗 𝒈 𝑳 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏 𝒎𝑳 𝒄𝒎 𝟑 𝟐.𝟓𝟒 𝟑 𝒄𝒎 𝟑 𝒊𝒏 𝟑 𝟏𝟐 𝟑 𝒊𝒏 𝟑 𝒇𝒕 𝟑 𝟏 𝒍𝒃 𝟒𝟓𝟑.𝟔 𝒈 =?𝒍𝒃𝒔 This is the calculation in dimensional analysis form (aka complete dimensional analysis form). As you work more problems, you might change the order of the conversion factors.

Density Problem - Using your calculator to avoid rounding errors and expessing your final answer with the correct number of SFs 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝟏.𝟏𝟗 𝒈 𝑳 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏 𝒎𝑳 𝒄𝒎 𝟑 𝟐.𝟓𝟒 𝟑 𝒄𝒎 𝟑 𝒊𝒏 𝟑 𝟏𝟐 𝟑 𝒊𝒏 𝟑 𝒇𝒕 𝟑 𝟏 𝒍𝒃 𝟒𝟓𝟑.𝟔 𝒈 =?𝒍𝒃𝒔 With the conversion factors rearranged, it is clear what units have canceled each other: 𝟏𝟓𝟓𝟎 𝒇𝒕 𝟑 𝟏𝟐 𝟑 𝒊𝒏 𝟑 𝒇𝒕 𝟑 𝟐.𝟓𝟒 𝟑 𝒄𝒎 𝟑 𝒊𝒏 𝟑 𝟏 𝒎𝑳 𝒄𝒎 𝟑 𝑳 𝟏𝟎𝟎𝟎 𝒎𝑳 𝟏.𝟏𝟗 𝒈 𝑳 𝟏 𝒍𝒃 𝟒𝟓𝟑.𝟔 𝒈 =?𝒍𝒃𝒔 You may then get the answer by, on your calculator, entering 1550 x 123 x x 1.19 / 1000 / = 115 lbs. Since we are allowed only 2 SFs, 120 lbs