Units of Measurement

Science is meaningless without mathematics.  We do calculations in science especially in chemistry and physics.  Calculations is also meaningless without units attached on it or number without unit is meaningless.

Science involved activities which needs to be measured by using measuring instrument like platform balance, graduated cylinder, ruler, etc.  Using this instruments have corresponding units to be used. Properties of matter also are mostly quantitative and they are associated also with numbers.  The properties can be meaningless without using units.  Like for example the density of water, when we just say 1, it doesn't have meaning; but when we say 1 g/mL, that is now understood that we are talking about density because of the attached unit.  Can you now see the difference?

In this post, I will be focusing on the metric system of measurement.  The metric system was first developed in France during the late 18th century,  and is used by most countries throughout the world.

SI BASE UNITS

In 1960, there was an agreement specifying the use of metric units in scientific measurement.  These units are called the SI Units , which means International System of Units. ( taken from the French Systeme International d' Unites). The SI system has seven SI Base Units and this is also the basis of some derived units.   See table below for SI Base Units:

We sometimes convert base units to decimals and multiples of various units, example we wanted to express kg to g or mg.  In this case prefixes can be used.  Table below will show different Prefixes with the corresponding values.

Mass and Weight

Mass and weight are often used interchangeably, but the two are different quantities.  Mass is a measure of the quantity of matter in an object, while weight is the force that gravity exerts in an object.  Weight is dependent on the location of an object while the mass does not.  The mass of the apple is constant and does not depend on its location but its weight will be different depending on its location.  Example, apple on the surface of the moon would weigh only one-sixth what it does on earth, simply because on the moon, it has less gravity.

The SI base unit of mass is kilogram (kg), which is equal to 1000 g.  In the laboratory, the most convenient way to use unit is g, because we measure small quantities of matter.  That's is why in calculation, once we need value in kg we do conversion of units using the prefixes in the table above.

Temperature

Temperature is another SI base quantity which has Kelvin as the SI unit.  It is a measure of hotness and coldness of a body. It can be measured using thermometer.  There are three temperature scales, there are degree Celsius (^oC), degree Fahrenheit (^oF), and the Kelvin (K) scale. 

Below are the comparison of the three scales:

The freezing point of water in degree Celsius is 0^oC, which is equivalent to 273.15 K in Kelvin, and  the boiling point of water in degree Celsius is 100^oC which is equivalent to 373.15 K in Kelvin.  Absolute zero in Kelvin is equivalent to -273.15^oC.  Both Celsius and Kelvin scales have the same magnitude.  As such, we can convert one scale to another by using the formulas below:

K  = ^oC  + 273

^oC  = K  -  273

In degree Fahrenheit and degree Celsius comparison on the other hand, is different.  That is why the formulas are different when converting one from the other.  The size of Fahrenheit scale is only 5/9 of a degree on the Celsius scale. Therefore the formula are as follows:

^oC = 5/9 (^oF - 32)

^oF = 9/5 (^oC) + 32 

Sample Exercise:

If a weather forecaster predicts that the temperature for the day will reach 31^oC, what is the predicted temperature in a) in K, and b) in ^oF?

Solution:

a)  K  = ^oC  + 273

          = 31  +  273

          =  304 K

b)  ^oF  =  9/5(^oC)  +  32

            =  9/5(31)  + 32

            =  56 +  32

            =  88 ^oF

DERIVED SI UNITS

Derived units are based from SI base units.  In chemistry we usually used volume, density, pressure, force, energy, etc,  as examples of derived units.  

Volume

Volume is length (m) cubed.  Thus the SI Derived unit for volume is cubic meter or m³.  Sometimes, smaller units such as cm³ and dm³ are also used in chemistry.

1 cm³ = 1 x 10^-6 m³

1dm³ = 1x 10^-3 m³

Another common unit for volume is liter (L), which is a volume occupied by one cubic decimeter (dm³).  One liter (L) is equal to 1000 ml or 1000 cubic centimeters.  One cubic centimeter is equal to 1 milliliter (mL).

                                                                  1L  = 1000 mL

                                                                         = 1000 cm³

                                                                         =  1 dm³

Density

Density is the amount of mass in a unit volume of a substance.

                                              

The SI derived unit for density is kilogram per cubic meter (kg/m³). This is large unit for chemical calculations and so gram per cubic centimeter (g/cm³) or gram per milliliter  (g/mL) are used instead. 

Below is a table for some densities of some selected substances at 25^oC:  

Sample Problem:

a) Gold is a precious metal that is chemically unreactive.    A piece of gold with a mass of 301 g has a volume of 15.6  mL.  What is the density of gold?

b)  Calculate the volume of 65.0 g of the liquid methanol if its density is 0.791 g/mL.

c)  A piece of platinum metal with a density of 21.5 g/mL has a volume of 4.49 mL.

Solution:

a) Density = mass / volume

                 = 301g / 15.6 mL

                 =  19.3 g/mL

b)  Volume = mass/density

                  = 65.0g / 0.791 g/mL

                  =  82. 2 mL

c)  Mass = volume x density

               =  4.49 mL x 21.5 g/mL

               =   96.5 g

Addition, Subtraction, Multiplication and Division of Scientific Notation

Very large and small numbers are difficult to deal with in calculation.  Let say for example you want to multiply a small number like 0.000 000 000 000 000 233 with a large number like 6,000,000,000,000,000,000.  Do you think you can multiply such numbers easily without converting those numbers in scientific notation?  For me, I don't think so.

To do such calculations easily we can change the number first to scientific notation and follow the rules in multiplying scientific notations.  Let' us have the rules in doing arithmetic calculations of scientific notation.

Adding and subtracting scientific notation

In adding and subtracting scientific notation, we then change the exponent to a common exponent.
Meaning the exponent of the two given  must have the same exponent.  Then, once the exponent is already the same you can also do the required operation.

Let us have an example:

1.  Add 24,000,000,000,000,000 to 5,000,000,000,000,000

We first change the numbers to scientific notation.

       2.4 x 10¹⁶
 +    5.0 x 10¹⁵

---

   
Since we cannot add immediately the two values because of their different exponents, we can change either of them  and still you will get the same result.  Let us try following the higher exponent 16 meaning we need to change the second scientific notation 15 to 16.  So, how do we do it?
We just move the decimal point once to the left the exponent 15 will become 16. Moving the decimal point to the left will increase the positive exponent and will cause the decrease of the negative exponent.

  2.4 x 10¹⁶
  0.5 x 10¹⁶

---

 2.9 x 10¹⁶

Just add the value of N and just copy x 10 and the common exponent.

How about if we will use the lower exponent, 15 as the common exponent?  To decrease the exponent from 16 to 15 we need to move the decimal point to the right once.  Just see the example below:

24.0 x 10¹⁵

+ 5.0 x 10¹⁵

---

 29.0 x 10¹⁵    or  2.9 x 10¹⁶

Since the value of N does not follow the standard form which should be not less than 1 and not more than 10 we need to move the decimal point again to follow the standard form of scientific notation.  We move the decimal point to the left and we increase the exponent to 16.  As you can see we still have the same answer.

Let us have an example of subtraction with a negative exponent.

2.    Subtract 0.000 000 000 000 000 345 from 0.000 000 000 000 002 4

Change the numbers first to scientific notation

  2.40 x 10^-15

- 3.45 x 10^-16

---

Before we calculate the problem above we need to change the exponent to a common exponent.  Let us use the higher exponent so that we do not move again the decimal point in the final answer.

   2.400  x 10^-15

-  0.345  x 10^-15

---

   2.055  x 10^-15

We did not move the decimal point since the answer is already expressed in correct form of scientific notation.

Multiplication and Division of Scientific Notation

In multiplying scientific notation, the value of N is multiplied and the exponent is being added.  You have to follow addition of signed number since the exponent, n, is either positive or negative integer.

Example:

                     (8.0 x 10⁴)(5.0 x 10²) = (8.0 x 5.0)(10⁴⁺²)
                                                         = 40 x 10⁶
                                                         = 4.0 x 10⁷

   
                  (4.0 x 10^-5)( 7.0 x 10³) = (4.0 x 7.0)(10^-5+3)
                                                        = 28 x 10^-2
                                                        = 2.8 x 10^-1


In dividing scientific notation the value of N is divided but the exponents are subtracted.  Also follow the rules in subtracting signed numbers.

Example:

                (6.9 x 10⁷) ÷  ( 3.0 x 10^-5)  =  (6.9 ÷ 3.0)(10^7-(-5)
                                                            =  2.3 x 10¹²


                (8.5 x 10⁴) ÷  (5.0 x 10⁹)  = (8.5 ÷ 5.0)(10^4-9)
                                                          = 1.7 x 10^-5

Scientific Notation

In science, we measure, calculate, round off numbers,  We encounter very small and large numbers which need to be calculated.  You will encounter problems if you do not know what is a scientific notation because even calculators when dealing with big and very small numbers convert the answers to scientific notation.  This post will help you how to write numbers in scientific notation and vice versa.


What is Scientific Notation?

Scientific notation is a shorter way of writing very large and small numbers.  It has a general form of

N x 10ⁿ 

where

               N  is a number which is not less than 1 and not more than 10

               n  is an exponent, it can be positive or negative integer (whole number)

Let us see for example we have 1 g of hydrogen, it contains this number of atoms

602,200,000,000,000,000,000,000

How to write number, in scientific notation form?

In expressing number in scientific notation, we have to locate the decimal point.  In a large number like the example above it shows no decimal point but it is fully understood that the decimal point is located at the last zero.  Now in setting the N we have to move the decimal point such that we can get a number that is in between 1 and 10.  And so we should move the decimal point to the left stopping in between 6 and 0, the number should be 6.02, then copy x 10.

Now in determining the value of n, we need to count the number of decimal places the point has moved.  From the last zero to in between 6 and zero it takes 23 decimal places and moving to the left and so the final scientific notation of the number above is

6.02  x  10²³

The exponent is positive since we moved the decimal point to the left which indicates that the number is a big number.

Let us have another example, the mass of 1 hydrogen atom which is

0.000 000 000 000 000 000 000 001 66 g

In this example, we set the value of N by moving the decimal point to  in between 1 and 6 to have 1.66 as the value of N.  To determine the exponent, we count the number of decimal places the point has moved.   It takes 24 decimal places from left to right which means that the exponent is negative that indicates that the number is very small number. Below is the scientific notation form of the number above:

1.66 x 10^-24 g 

In summary, we can write very small and large number in  scientific notation in two ways:

1.  Just set the value of N by moving the decimal point such that you can have a number in between 1 and 10.

2.  In determining the value of n, or the exponent, we just count the number of decimal places the point has moved.  The number of decimal places is the exponent.   Once the decimal point is moved from right to left, the exponent is positive and when move from left to right the exponent is negative.

More examples:

Write the scientific notation of the following numbers:

a.  0.000 000 005 5  

b.  0.000 000 000 000 430

c.   2,000,000

d.   400,000 000 000

Solution:

a.  5.5 x 10^-9

b.  4.30 x 10^-13

c.  2.0 x 10⁶

d. 4.0 x 10¹¹

How to convert  scientific notation to ordinary number?

Here are simple ways to convert scientific notation to ordinary number:
1.  Identify if small number or big number by looking at the exponent.  If the exponent is positive; it means its a big number if negative; its a small number.  Once positive we need to add zeros to the right and once negative we will add zeros to the left.

2.  Add zeros by also counting the decimal places equivalent to the exponent starting from the decimal place where the decimal point is located.

Example:

1.  3.55 x 10⁶  

     The exponent is a positive integer, meaning it's a big number; therefore we need to add zeros to the right.  It doesn't mean you will add 6 zeros but you have to count starting from the decimal place in between 3 and 5 where the decimal point is located.  Moving to the right adding zeros until you reach 6 decimal places.  And so the answer is 3,550,000.

2.  6.23 x 10^-9

      The exponent above is negative integer, which means the number is very small number; and so we need to add zeros to the left. In adding zeros you need to count the number of decimal places the decimal point is moved following the exponent.  The answer for number two example is
0.000 000 006 23.

 I hope you learn something from this post. Just try below:


TRY THIS:

1.  Write the numbers below in scientific notation:
     a.  345,000,000,000
     b.  5,000,000,000,000,000
     c.  0.000 000 000 000 000 000 4
     d.  0 000 567
     e.  0.000 000 008 5


2.  Convert the scientific notation below to ordinary number
     a.  8.5 x 10^-7
     b.  1.23 x 10^-12
     c.  4.30 x 10⁵
     d.  3.33 x 10⁹
     e.  2.9 x 10^-15

For addition, subtraction, multiplication and division of scientific notation click HERE.