Mass and Volume

After completing this unit you should be able to:
• convert between grams (g) and other units of mass
• compute answers with the correct precision
• convert between liters (L) and other units of volume

Mass

A simple definition of chemistry is that it is the scientific study of matter, it's properties, the changes it undergoes, and the energy associated with matter and it's changes.  So, what is matter?  A practical definition is that matter is anything that has mass.  Newton's second law defines mass as the property of matter that determines the force required to impart a given acceleration to an object:

force (F)     =     mass (m)     x     acceleration (a)

Mass describes the quantity of matter in a sample, and the sum of the masses of the components of a sample is equal to the mass of the whole sample.  The familiar unit of quantity in the US, pound (lb), is a unit of weight, not mass.  Weight is the force exerted by a mass under the influence of gravity:

weight (W)    =    mass (m)    x    gravitational acceleration (g)

The mass of a particular object is a fixed quantity, but acceleration due to gravity, and therefore weight, varies with location.  For our purposes we will assume that when we refer to a weight in pounds we mean as measured at a point on the Earth's surface where the gravitational acceleration is 9.80665 m/s2, under which conditions the unit can be read as pounds-mass.

The standard of mass in science is the kilogram (kg), which is the only SI unit still defined in terms of a physical object, a metal cylinder stored at the BIPM in Paris.  Copies of the standard kilogram are distributed to national laboratories around the world (the US copy is at NIST) and these are used to calibrate additional standards. The prefix kilo means thousand, so 1 kg = 1000 g. A single gram is a rather small unit of mass. For reference, a US five-cent coin (a nickel) has a mass of 5.000 g.

The pound is defined by reference to the gram, and as with some of the other definitions we have seen, the number is chosen to give good agreement with traditional values.  A conversion factor with four significant figures will be satisfactory for most purposes.  You should memorize the conversion factor with 2 sf. so that you can do quick estimates of magnitude when you are given a value in a metric unit.

 US System SI System 1 short ton ≡ 2,000 pounds (lb)* 1 lb ≡ 16 ounces 1 lb ≡ 7,000 grains* 1 Mg ≡ 1 x 106 g = 1 metric ton 1 kg ≡ 1 x 103 g 1 mg ≡ 1 x 10-3 g 1 μg ≡ 1 x 10-6 g 1 lb ≡ 453.59237 g* = 453.6 g (to 4 sf.) 1 kg = 2.205 lb (to 4 sf.)* = 2.2 lb (to 2 sf.)

Problem:  After spending a month panning for gold you find you have 12.3 ounces.  You look up the price of gold on the internet and find a value of \$42.16 per gram.  How much is your gold worth?

Solution:  Our measured quantity is in ounces, but the price is given per gram, so we start by converting the units of our quantity from ounces to grams.  We don't know a direct conversion factor, but we can chain several factors together, checking with each factor that the units cancel correctly:

The price gives us another definition for a conversion factor: for gold today, 1 g = 42.16 dollars, and we use this to calculate the final answer to the problem:

The final answer is rounded to 3 sf. because the original quantity had three significant figures, and all of the conversion factors used have 4 sf. or are exactly defined.

Precision in Calculated Values

We have seen that when calculations involve multiplication and division, the precision of the answer is represented by rounding to the same number of significant figures as there are in the measurement with the smallest number of sf.  We will now see that addition and subtraction must be treated differently.

Suppose we have 1.03 g of table salt, and add to it 27 mg more salt.  How much salt do we have now?  Before we can add the two values together we must convert one of the quantities to the same units as the other quantity.  Addition and subtraction require that values have the same units.  It doesn't matter which value we convert.  Let's choose to convert 27 mg to grams:

We also need to express the values in the same form, so that the decimal point has the same meaning in both values.  In this case we have chosen not to use scientific notation.  Now we can add the two values together, and round off to the same last place as the measurement with the largest uncertainty.

Notice that we knew the larger mass only to the nearest 0.01 g (i.e. hundredth of a gram).  Even though we knew the smaller mass to the nearest 1 mg or 0.001 g (thousandth of a gram), we can't know the total any better than to the nearest 0.01 g.  Notice also that even though the value of the smaller mass has only 2 sf, our answer correctly has three significant figures.

When the numbers to be added or subtracted are written in a stack with the decimal points aligned, then the last significant figure in the answer is in the rightmost column where every value has a significant digit:

1.03
0.027
1.057

Even when measurements have the same units, if the values are expressed in scientific notation we can only add or subtract the coefficients when the exponents are the same.  For example, to add 9.83 x 104 g and 8.24 x 103 g we must first change the form of one value so that it has the same exponent as the other value.  Then we can add the coefficients, round off, and express the final answer in scientific notation.

It is very important to remember that calculators do not understand precision. Simply entering the two numbers in exponential form and taking the sum will give the answer as 106,540 or 1.0654 x 105 depending on how your calculator is set up. Answers read from the display will usually need to be rounded or have trailing zeroes added, whichever is appropriate, based on your analysis of the calculation.

Volume

Another property of matter is that it occupies space, i.e. it has volume.  The volume of a rectangular block is found by multiplying together the lengths of the sides.  A cube 6.00 cm on each side, for example, has a volume of 6.00 cm x 6.00 cm x 6.00 cm = 216 cm3.  Notice that the unit of volume is length cubed.

For a rectangle, length x width = area.  The units of area are length squared, and volume = area x height.

The volume of a cylinder with a diameter of 2.7 cm and a height of 18.3 cm is found as follows:

Notice that the least precise measurement, the diameter, has two significant figures, so the answer is rounded to two significant figures and expressed in scientific notation to avoid ambiguity.

Pi (π, the circumference of a circle divided by it's diameter) is an irrational number.  Unlike a rational number, it cannot be expressed exactly as a ratio of integers, and its decimal representation (3.14159...) continues for ever without recurring.  In calculations, we choose a value of π that is more precise than our measurements so that it does not limit the precision of our answer.

Volume is length cubed, so to convert among volume units we use the same table of conversion factors discussed for length, but the values and units must be cubed.

For example, a volume of 62.5 cubic feet can be converted to cubic meters as follows:

Alternatively we can start by cubing the defining relationship.  For example, to express this answer in cm3:

Familiar units of volume, gallons and pints for example, are not used in science.  If we want to use a unit like these, we can use the liter (L)One liter is defined as the volume of a cube 1 dm on a side.

 US System SI System 1 gallon ≡ 4 quarts 1 quart ≡ 2 pints 1 pint ≡ 16 fluid ounces 1 L ≡ 1 dm3 1 L = 1000 cm3 1 mL = 1 cm3 1 μL ≡ 1 x 10-6 L 1 L = 1.057 quarts (to 4 sf.)* = 1.1 quarts (to 2 sf.)

The relationship between US units and SI units of volume is measured, not defined, and a value to 4 sf. is sufficient for most purposes.