Graphing and Density
In addition to values that you can directly measure like length, mass, or volume, chemistry is
filled with values that must be calculated. The most common of these is density. Density is
defined as the mass of an object per unit volume. To put it more commonly, density is how
“heavy” or “light” something is. We use the quotation marks on those words because heavy
and light refer to mass but if you have a substance that is very dense, it feels heavy for a small
object. Something that is not very dense has a very small mass for a big object.
Let’s put it another way: what weighs more, a pound of feathers or a pound of bricks? The answer, of course, is neither. They both weigh a pound so their mass is equal. However, imagine
how many bricks you’d have to get to amass a pound; probably only one. Think of how many
feathers it would take to get a pound; that’s a lot of feathers. The brick is very dense– it has a
lot of mass in a small space. The feathers are not dense– they have little mass in a lot of space.
1 pound of brick
1 pound of feathers
Mathematically, density goes by the following formula:
Density = mass
volume
You can use this equation to solve for density, mass, or volume as long as you know two of the
three.
Examples:
A) What is the density of a substance that has a mass of 100 grams and volume of 20 mL?
density = X
mass = 100 grams
volume = 20 mL
D=
m
v
X = 100 g
20 mL
X = 5 g/mL
B) What mass is 500 mL of a substance that has a density of 14 g/mL?
density = 14 g/mL
mass = X
volume = 500 mL
D=
m
v
14 g/mL = X
500 mL
X = (14 g/mL)(500 mL)
X = 7000 g
C) How much space does 25 grams of a substance with a density of 3 g/mL occupy?
density = 3 g/mL
mass = 25 g
volume = X mL
D=
m
v
3 g/mL = 25 g
X
X = 25 g
3 g/mL
X = 8.33 mL
Measuring Volume
Density is a property of matter that cannot be directly measured easily. One must measure both
the mass and volume of a substance for this to occur. Measuring the mass of a substance is trivial if you have a digital scale. However, measuring the volume of a substance can be very
tricky, especially if the substance is irregular in shape. If the substance is a cube or other regular shape, you can use formulas like V = Length x Width x Height to calculate the volume but if
the substance is a lump, how would you measure the amount of space it occupies? One of the
easiest methods to use is that of Archimedes’ principle of water displacement. The idea is as
follows: if a known quantity of water is measured for its volume and a substance is placed into
the container, the substance will displace some of the water, causing its volume to increase.
The amount of volume the water increased, is the amount of volume the substance occupies.
The graduated
cylinder has 20
mL of water in
it here.
http://www.juliantrubin.com/encyclopedia/
physics/buoyancy_files/Submerged-andDisplacing.png
Archimedes of
Syracuse lived
approximately
from 287 to 212
BC. He is considered the greatest mathematician of the Ancient World.
When the stone
is put in, the volume increases to
23 mL. Thus the
stone has a volume of 23-20 or
3 mL.
http://www.fredsakademiet.dk/
library/science/_gifs/
archimedes.jpg
Density and Graphing
Once you measure the mass and volume of a substance, calculating the density should be easy.
There is, however, another way to represent density and that is by graphing it. Density is the
relationship between mass and volume so our graph will reflect that.
Volume (ml) Mass (g)
10
50
20
100
30
150
40
200
50
250
If this graph is a true reflection of density, where is the density? We see the mass, we see the
volume, but where is the density? Remember that D = m/V. How are we going to get that?
Look at the relationship: a normal straight line is y = mx +b. We can see that the y-intercept is
at 0,0 so our equation becomes simpler: y = mx. Let’s substitute in our variables for y and x.
On the y-axis we have mass and on the x-axis we have volume so:
y = mx
Mass = m (volume)
Rearranging the equation:
m = Mass
Volume
What is “m” in a straight line equation? It is the slope of the line. From the equation above:
y = 5x
the slope of the line is 5 so the density of the substance is 5 g/mL
Comparing Densities Graphically
If you compare different substances on the same graph, you can tell which are more dense relative to each other. Here is a set of data for three different substances and their masses at various
volumes:
Volume (mL) Mass A (g) Mass B (g) Mass C (g)
0
0
0
0
10
2
12
8
20
4
24
16
30
6
36
24
40
8
48
32
50
10
60
40
Densities of Various Substances
70
60
y = 1.2x
50
Mass A (g)
Mass (g)
40
Mass B (g)
Mass C (g)
y = 0.8x
Linear (Mass A (g))
30
Linear (Mass C (g))
Linear (Mass B (g))
20
10
y = 0.2x
0
0
10
20
30
40
50
60
Volum e (m L)
You can tell by the steepness of the lines that substance B is the most dense and substance A is
the least dense. Looking at the slope equations, their densities are as follows:
A = 0.2 g/mL B = 1.2 g/mL C = 0.8 g/mL
Water has a density of 1 g/mL. Which of these substances would sink in water and which
would float? A substance will sink in water if it is more dense and will float if it is less dense.
If we draw in a line for water:
Densities of Various Substances
70
60
y = 1.2x
H2O
50
Mass A (g)
Mass B (g)
Mass C (g)
y = 0.8x
Linear (Mass A (g))
30
Linear (Mass C (g))
Linear (Mass B (g))
20
10
y = 0.2x
0
0
10
20
30
40
50
60
Volum e (m L)
We can see that substance B is more dense than water so will sink. Substances A and C are less
dense than water so will float.
http://www.stevespanglerscience.com/img/
cache/
bcb9b8db117ee64376aedaf7af3595ca/
sevenlayer-2-51908.jpg
Mass (g)
40
This picture shows four different liquids in what is
called a density column. The bottom layer is the most
dense and the top layer is the least dense.
Questions
http://www.cstephenmurray.com/
onlinequizes/chemistry/measuring/
Dinocopy.jpg
1. What is the formula for density?
2. An object has a mass of 50 grams and a volume of 20 mL. What is its density?
3. An object has a mass of 70 grams and a density of 0.8 g/mL. How many mL does it occupy?
4. An object has a volume of 125 mL and a density of 3 g/mL. What is its mass?
5. A graduated cylinder was filled to the 20 mL line. Then some stones were put in and the
water level rose to 27 mL. What is the volume of the stones?
6. A plastic dinosaur has a mass of 7.5 grams. With this information and the picture below,
what is the density of the dino?
7. Graph the following data below. Calculate the density of the object. Would the object sink
or float in water? Why?
Density
20
Volume (mL) Mass (g)
0
0
5
3
10
6
15
9
20
12
25
15
Mass (g)
15
10
5
0
0
5
10
15
Volum e (m L)
20
25
30