Density

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Discussion

Which is heavier: a kilogram of feathers or a kilogram of lead?

Formally, density is the ratio of mass to volume for a material

ρ =  m
V

Units

1000 kg/m3 = 1000 g/l = 1 g/cm3

liquids are typically hundreds of times more massive than gases.

Densities of Selected Materials (~20 °C, 1 atm)
material density
(kg/m3)		   material density
(kg/m3)
acetone 790   mayonnaise, traditional 910
acid, acetic (CH3COOH) 1050   mayonnaise, light 1000
acid, hydrochloric (HCl) ????   methane, gas, +25 °C 0.656
acid, sulfuric (H2SO4) 1390   methane, liquid, -90 °C 162
air, 100 K 3.556   milk, cow, heavy cream 994
air, 200 K 1.746   milk, cow, light cream 1012
air, 293 K 1.207   milk, cow, whole 1030
air, 300 K 1.161   milk, cow, skim 1033
air, 500 K 0.696   mercury 13,594
air, 1000 K 0.340   monosodium glutamate 1620
alcohol, ethyl (grain) 789.2   nickel 8900
alcohol, isopropyl (rubbing) 785.4   nitrogen (N2), gas, ~300 K 1.145
alcohol, methyl (wood) 791.3   nitrogen (N2), liquid, 74 K 808
ammonia 771   oil, vegetable, coconut 924
aluminum 2700   oil, vegetable, corn 922
argon, gas, ~300 K 1.449   oil, vegetable, olive 918
argon, liquid, 87 K 1430   oil, vegetable, palm 915
beer, pilsner, 4 °C 1008   oil, vegetable, peanut 914
benzene 870   oil, vegetable, soya 927
blood 1035   osmium 22,500
body fat 918   oxygen (O2), gas, ~300 K 1.308
bone 1900   oxygen (O2), liquid, 87 K 1155
butane 551   perchlorethylene 1600
butter 911   platinum 21,450
carbon 2250   plutonium, α 19,860
carbon dioxide, gas, +25 °C 1.799   salt (sodium chloride) 2165
carbon dioxide, solid, −37 °C 1101   silicon 2330
copper 8960   silicon dioxide (quartz) 2600
corn starch, loosely packed 540   silicone 993
corn starch, tightly packed 630   silver 10,490
corn syrup 1380   skin 1050
diesel 800   sodium bicarbonate 2200
formaldehyde 1130   sugar, sucrose 1550
freon 12, liquid 1311   titanium 4500
freon 12, vapor 36.83   tungsten 19,300
gasoline 803   uranium 19,050
glycerine 1260   water, liquid, 100 °C 958.40
gold 19,300   water, liquid, 50 °C 988.03
helium, gas, ~300 K 0.164   water, liquid, 30 °C 995.65
helium, liquid, 4 K 147   water, liquid, 20 °C 998.21
hydrogen (H2), gas, 300 K 0.082   water, liquid, 10 °C 999.70
hydrogen (H2), liquid, 17 K 71   water, liquid, 4 °C 999.98
honey 1420   water, liquid, 0 °C 999.84
iron 7870   water, ice, 0 °C 916
iridium 22,400   water, ice, -50 °C 922
kerosene 810   water, ice, -100 °C 927
lard 919   water, sea 1025
lead 11,350   zinc 7140
lithium 534      
lithium 6 deuteride 820      
lungs 400      

Summary

Problems

practice

  1. A lucite cube has a mass of 142.5 g and a width of 4.9 cm. Determine its density in kg/m3.
         
     
         

    Solution …

    Start with the definition of density, replace V with s3 (the volume of a cube), substitute the given quantities in SI units, and solve.

    ρ =  m  =  m  =  (0.1425 kg)  = 1200 kg/m3
    V s3 (0.049 m)3
  2. It has been known for several thousand years that the earth is spherical (by educated people, at least). Sometime in the Second Century BCE the size of the earth was determined (rearth = 6,370 km). By the Nineteenth Century its mass was known (mearth = 5.97 × 1024 kg). And in the early Twentieth Century the structure of the earth was deduced. The earth has three main layers: crust, mantle, and core. The crust of the is the lightest and thinnest and, like the shell of an egg, contributes little to its overall mass. The mantle is a bit more dense, substantially thicker, and contains most of the earth's mass. The core is the densest layer (but not the most massive) and is divided into a liquid outer core and a solid inner core. The relevant data for the interior of the earth are summarized below.
     
    layer depth range (km) mean density (kg/m3) consistency
    crust 0 ~ 20 2700 solid
    mantle 20 ~ 2890 4500 plastic
    outer core 2890 ~ 5160 ? liquid
    inner core 5160 ~ 6370 ? solid
     
    Determine …
    1. the average density of the entire earth
    2. the percent of the earth's mass located in the mantle, and
    3. the average density of the core.

    Solutions …

    1. Density is the ratio of mass to volume. Use the given mass of the earth and calculate its volume from the radius and the formula for the volume of a sphere.
                   
      ρearth =  mearth  =  3m  =  3(5.97 × 1024 kg)  = 5514.01 … kg/m3
      Vearth 4πr3earth 4π(6.37 × 106 m)3
      ρearth =  5510 kg/m3
                   
    2. You may want to do this is stages. Calculate the volume of the spherical shell that is the mantle.
                   
      Vmantle =  4  πr3outer −  4  πr3inner =  4  π ( r3outer − r3inner )
      3 3 3
      Vmantle =  4  π [ (6,370,000 m)3 − (3,480,000 m)3 ]
      3
      Vmantle =  9.06164 … × 1020 m3
                   
      Mass is density times volume. Use the density given in the data table and the mass calculated above.
         
      mmantle =  ρV = (4500 kg/m3)(9.06164 … × 1020 m3)
      mmantle =  4.07774 … × 1024 kg
         
      Divide the mass of the mantle by the mass of the earth to determine the percentage of earth's mass in its most substantial layer.
             
      mmantle  =  4.07774 … × 1024 kg  = 0.683038 … ≈ 70%
      mearth 5.97 × 1024 kg
             
    3. This part could also be solved in stages, but let's try a different approach. Start from the basic definition of density, replace mass and volume with their respective fractions of the earth's total, then multiply by the average density. Recall that the volume of a sphere is proportional to the cube of its radius.
         
      ρcore =  mcore  =  (1 − 0.683038 …)mearth  = 1.94396 … ρearth
      Vcore ⎛
      ⎝      		 6370 km − 2890 km ⎞3
      ⎠      		 Vearth
            6370 km  
      ρcore =  (1.94396 …)(5514.01 … kg/m3) = 10,719.02 … kg/m3
      ρcore =  10,700 kg/m3
         
      This number seems a bit low to me. The earth's core is homogeneous chemically, but not physically. Both layers are a mixture of iron and nickel, but one is liquid and the other solid. The liquid outer core is reported to range in density from 10,000 to 12,000 kg/m3 and the solid inner core from 12,300 to 12,600 kg/m3. My suspicion is that the mean value is more like 10,900 kg/m3, but I don't see any error in the solution presented above. I leave you with the classic excuse found in many college math textbooks, but with a slight modification. "The disproof is left to the reader."
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. Mayonnaise is essentially a mixture of vegetable oil and water with a bit of egg yolk added as an emulsifier (a substance that keeps the oil and water from separating). Traditional mayonnaise has a density of about 910 kg/m3 while reduced fat, low calorie, or "light" mayonnaise has a density of about 1000 kg/m3. What do these densities tell you about the composition of light mayonnaise?
  2. Why does "heavy cream" have a lower density than "light cream"? Explain this apparent contradiction.

numerical

  1. A room is 16.40 m long by 4.5 m wide by 3.26 m high. The density of air is 1.29 kg/m3. Find the mass of the air contained in the room.
  2. Determine the density of the following astronomical objects arranged in order of increasing mass. (Pay attention to the units.)
     
    object mass radius density (kg/m3)
    the sun 1.99 × 1030 kg 696,000 km
    white dwarf star 0.5 to 1.4
    solar masses    		 5000 km
    neutron star 1.4 to 3
    solar masses    		 10 km
    stellar black hole more than 3 solar masses 2Gm/c2 (event horizon)
    supermassive black hole > 106 solar masses 2Gm/c2 (event horizon)
    the known universe 1053 kg 13.7 × 109 light years
     
  3. Complete the following table with values in SI units.
     
    object mass (kg) radius (m) density (kg/m3)
    iron shot put 7.26
    (16 lbs)    		 7870
    iron atom
    (55.847 u)    		 7870
    iron atom's
    electron cloud
    (26 melectron)
    iron nucleus
    (55.847 u)
    (~4 fm)
     

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