                                        Continuing with units of measure - density and heat capacity

I. Density

"Which weighs more - a pound of feathers or a pound of lead?"

Feathers have more volume, but a pound is still a pound.

Mass is an extrinsic property in that it depends on how much we have.  Better for comparison are instrinsic properties that depend only on what the material is.

Density - physical property that relates the mass of an object to volume.

 density = mass volume

typical unit (non-SI) -  g / cm3  = g cm-3 = g / ml = g ml-1

 Material Density (g / ml) Ice (0oC) 0.917 Water (3.98oC) 1.00000 Water (20oC) - close to RT 0.997 Gold 19.3 Helium 0.000194 Air 0.0012 Human Fat 0.94 Earth 5.54

DEMO: Comparing densities of gases  Halocarbon 4.15 g / L, O2 - 1.3 g/L, H2 0.081 g / L, He 0.2 g /L  (note unit change)

Many are temperature dependent - in particular, the density of a gas changes a lot with temperature whereas the density of solids and liquids to not change as sharply.

DEMO: show w/ gas in LN2

"Helium is lighter than air" - We really mean helium is less dense then air.

Note: Ice is less dense than water, ice floats - very important to aquatic life
unusual - few materials are less dense in the solid form.  Generally, the density of the solid is greater than density of the liquid is much greater than the density of the gas.

Using density:

Density is used to characterize substances and to convert between volume and mass.  For measuring out solids mass is generally more convenient, for measuring out liquids, volume is generally more convenient.

Sample Problem: What volume of isopropyl alcohol (rubbing alcohol) would you use if you needed 25g?  Information:  D = 0.7855 g / cm3 at 20oC

 D = m V

Rearranging yields:

 V = m 25g cm3 = 31.8 cm3 D 0.7855g

Does this make sense?  Density less than one so the volume in mL (cm3) is greater than the mass in grams.

II. Specific gravity - often it is more convenient to measure the density of a substance relative to that of water

 specific gravity = density of substance (g / mL) density of water at same temp (g/mL)

As it is a relative measure, it has no units, they divide out.
Independent of temperature, water has a specific gravity of 1.0
Hydrometer - means of measuring specific gravity - more it sinks the higher the specific gravity Where do we use hydrometers?

winemaking - monitoring the conversion of sugars to alcohol (ethanol) during fermentation

 Substance Specific gravity Grape Juice - 20% dissolved sugar 1.082 Dry wine - 12% alcohol 0.984 pure alcohol (ethanol) 0.789

urine analysis - normal sg or uring 1.003 - 1.030.   Abnormalities can indicate diabetes

diabetes mellitus (what many normally associate with the word diabetes) - high reading due to excess dissolved sugar (glucose) in the urine.

diabetes insipidus - low specific gravity, nearly colorless urine, very little dissolved material.

III. Heat capacity

Remember heat measured in Joules, calories (1 kcal = dietary Calorie)

1 cal = 4.18 J

Heat capacity (symbolized Cp, units: cal / oC, J / oC, etc... ) - quantity of heat required to change an objects temperature by 1oC.  Depends on both the mass of an object (so it is an extrinsic property) and its composition.

The air contained in an over has a lower heat capacity than the metal racks.  The metal racks can store more "heat" than the air.  More energy as heat is transferred to your hand when you touch the rack than simply place your hand in the air of the oven.

Specific heat capacity (units: cal g-1 oC-1, J g-1 oC-1, etc... ) - not relative to something as with specific gravity.

quantity of heat required to raise the temperature of 1 gram of a substance 1oC.  Takes into account weight so it is a property characterizing a material; it is an intrinsic property.  Often better for making comparisons between materials.

 Substance Specific Heat [cal / (oC g)] water (by definition of calorie) 1.00 glass 0.12 iron 0.11

The high specific heat of water means it can store a lot of energy (passive solar energy schemes - large drums of water)!  Good at removing heat from other objects (sweat, nuclear reactors)

 specific heat (cal / (g oC))= heat (cal) mass(g)  change in temp (oC)

In symbols,

 specific heat (cal / (g oC))= H m DT

D = change

Rearranging yields:

heat = m DT s.h.

where m is mass, DT is the change in temperature, and s.h. is the specific heat capacity

Problem:  Taking a bath might use 95kg of water.  How much energy (in cal and Joules) is needed to heat the water from 15oC to 40oC.   Remember, specific heat of water is 1.0 cal / (g oC)

Change in temperature = DT = 40-15 = 25oC

Does it matter if we are in oC or K or oF?

mass = 95 kg = =95000 g = 9.5 x104 g

heat = (1.0 cal / (g oC))(25oC)(9.5x104g) = 2.375 x 106 cal

heat = (2.375 x 106 cal) (4.18 J / cal) = 9.927 x 105J = 9.9 x 105 J

A more compact approach: We can treat specific heat much like a conversion factor.  Although unlike converting between pounds and kg, specific heat converts between three quantities (mass, heat, temperature change).  If we know two, we can determine the third.  As we can treat specific heat like a conversion factor, we can carry out the calculation much as we would carry out a unit convertion problem:

 9.5 kg 1000g 1.00 cal 25oC 4.18J = 9.9 x 105 J 1kg g oC 1 cal

How much heat is this?

e.g. combustion (exothermic process) - ideally: carbon compound + oxygen -> carbon dioxide + water
 wood (1kg) 4,300 kcal = 4.3 x 106 cal coal (1 kg) 7,400 kcal Gasoline (1 kg) 11,500 kcal Nuclear reaction (1 kg) - not combustion 20,000,000,000 = 2x1010 kcal