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Dimensional Analysis Notes

The Basic Units of Measurement

  • International System of Units (SI)
  • SI Standard Units
    • length - meter - m
    • mass - kilogram - kg (it is the only one that uses a prefix)
    • time - second - s
    • temperature - kelvin - K
    • amount of substance - mole - mol (this is very important to use - it cannot be just “m” because you have the meter and the mass)
    • Luminous intensity will not be used in class and that would be a candella
    • We will use SI and non SI units because sometime you have to convert
    • Volume is not there and it is called an SI derived unit so it is not up there but it is very important (you can calculate it so you do not have to have it as a base unit)
    • mass - kg
    • mass vs weight
    • this one is an exception because it has a prefix
    • Mass is the amount of substance in an object and it changes the same regardless of where I go
    • Weight is resistance of mass due to gravity so if you go the moon, your weight decreases but your mass stays the same.
    • On earth, we can use them interchangeable because the gravity does not change so on earth they actually mean the same thing
  • Temperature Scale
    • Celsius scale
    • Based on water
    • We will use C a lot in lab but the SI unit is actually K
    • You can have negative C
    • Kelvin
      • absolute 0 (this is the point when we have no molecular motion and in theory we cannot ever reach it because we would not have any life).
      • This is good for use in equations because you can only have positive numbers
      • C → K + 273
      • K → C - 273

SI Prefixes

  • giga (G) - 1 x 10 9 greater - 1 Gm = 1 x 10 9 m
    • You can use any units here, you could use non SI units, or derived units or really anything you want to use with these prefixes
    • One gigameter would be a billion meters
    • One gigagram would be a billion grams
    • One gigamole would be a billion moles
    • mega(M) - 1 x 10 6 greater - 1 Mm = 1 x 10 6 m
    • kilo(k) - 1000 times greater - 1km = 1000m
    • You can use any units here, you could use non SI units, or derived units or really anything you want to use with these prefixes
    • hecto(h) - 100 times greater - 1hm = 100m
    • deka(dk, dc, da) - 10 times greater - 1dkm = 10m
    • Base unit (m, g, s, L, etc.). Any unit at all would work here because all of these prefixes work with anything so feel free to use them. Just be sure to use the same units in the same problem or you might run into issues. deci(d) - 0.1 smaller - 1m = 10dm
    • you can always write 1dm = .1dm but it is not preferential to write it this way because the odds are increased that you make a mistake
    • centi© - 0.01 smaller - 1m = 100cm
    • milli(m) - 0.001 smaller - 1m = 1000 mm
    • micro(insert micro symbol here [u]) - 1 x 10 -6 smaller - 1m = 1 x 106 um
    • nano (n) - 1 x 10 -9 smaller - 1m = 1 x 10 9 nm
    • pico(p) - 1 x 10 -12 smaller - 1m = 1 x 10 12 pm
  • Volume
    • derived unit
    • the typical units for this are L or mL
    • V = l x w x h for a cube or a rectangle of sorts
    • 1m 3 = 1000000cm 3
    • 1 m 3 = 1000 dm 3
    • 1 dm 3 = 1000 cm 3
    • 1 L = 1 dm 3
    • 1mL = 1 cm 3
  • Problem Solving and Unit Conversions
    • conversion factors
    • 1m = 100cm
      • a) 1m/100cm
      • b) 100cm/1m
    • Dimensional analysis
      • a conversion factor is equal to 0 and they are ratios
      • There are also two ways to write conversion factors
  • Simple Dimensional Analysis Problems
    • 0.000 0732 micrometers to meters
    • 0.000 0732 um * 1m/10 6 um = 7.32 x 10 -11 m
    • 83 grams to centigrams
    • 83g * (100cg/1g) = 8300 cg
    • 35 hours to seconds
    • 35h * 3600 sec/1h = 126 000 sec but we would write 130000 sec because of the significant figure rules
    • 4.56 L to cm 3
    • 4.56L * 1000 cm 3/1L = 4560 cm 3
    • 582 milliseconds to seconds
    • 582 ms * 1 sec/1000 ms = .582 ms
    • 0.002 84 hm to meters
    • 0.002 84 hm* 100m/1hm = .284m
    • 0.002 84 mL to dm 3
    • 0.00284 mL * 1 cm 3/1mL * 1 dm 3/1000cm 3 = .00000284 dm 3
  • Complex Dimensional Analysis:
    • 0.000 234 dm to nm
      • 1m = 10 dm
      • 1m = 10 9 nm
      • 0.000 234 dm * 1m/10dm * 10 9 nm/1m = 23400 nm
    • 2.62 x 10 -3 kg to mg
    • 2.62 x 10 -3 kg * 1000g/1kg * 1000mg/1g = 2620mg
    • 11.4 g/cm 3 to kg/m3
    • 11.4 g/cm 3 = 1kg/1000g * 1000000cm 3/1m 3 = 11400 kg/m 3
    • 56.8 m/s to km/hr
      • 56.8 m/s * 1km/1000m * 3600s/1hr = 204.48 km/hr (204 km/hr)
      • 0.0927 um to cm
    • 0.0927 um * 1m/1000000um * 100cm/1m = 9.27 x 10 -6 cm
    • 12.3 hs to ms
      • 12.3 hs * 100s/1hs * 1000ms/1s = 1230000ms
    • 0.0549 mL/min to nL/s
      • 0.0549 mL/min * 1L/1000mL * 1 x 10 9 nL/1L * 1min/60s = 915 nL/s
  • Units Raised to a Power
    • Problems
      • A square has an area of 345.5 cm 2. What is its area in square meters
        • 0.03455 m 2
      • A boat engine has a displacement of 4255 cm 3. What is its displacement in cubic inches? (1 inch = 2.54 cm)
        • 4255 cm 3 * 1 in 3/2.54 3 cm 3 = 259.7 in 3
  • Density
    • density = mass/volume
    • d = m/v
    • density of solids/liquids
    • density of gases
    • effects of temperature
    • volumes of liquids and gasses do not change with temperature
    • volumes of gasses do change with temperature
    • if you change the volume of a gas, you change its density
    • If you decrease the volume of a gas, you decrease its temperature
    • If you increase the temperature of a gas, you increase its volume
      • Problems:
        • A piece of metal has a mass of 57.3g and a volume of 4.75 cm 3, what is the density
        • d = m/v = 57.3g/4.73 cm 3 = 12.1g/cm 3
        • The density of lead is 11.4g/cm 3. If a piece of lead has a volume of 4.14 cm 3, what is the mass?
        • d = m/v 11/4g/cm 3 = x/4.14 cm 3 = 47.2 g
        • If CO2 has a density of 1.83 g/L and has a mass of 5.34 grams, what is the volume?
        • d = m/v 1.83g/L = 5.34g/x x1.83g/L = 5.34g x = 2.92L

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