# UNIT CONVERSIONS AND DIMENSIONAL ANALYSIS

**Unit conversion and Dimensional analysis **

**in general chemistry**

This deals with conversion of one unit of measurement to another unit in general chemistry. One unit of measurement can have several conversion units. For example, the **mass** of substance can be measured in units of grams (g). However, there are other units such as the milligrams(mg) , kilograms (kg) or the pounds( lb, lbm) . Another unit is distance or length. This can be measured in many units such as meters(m) , millimeters (mm), centimeters (cm) , inches, yards, kilometers (km) or miles.

In each conversion, it is important to know the prefix needed and how to use it to make a conversion factor or ratio. An example is given in this illustration below.

**LENGTH**

SI unit for length is meter.

**100centimeters = 1 meter**

**100cm = 1m implies 1cm= (1/100) m = 0.01m = 1.0 x 10 ^{-2}m **

**1cm=0.01m = 1.0 x 10 ^{-2}m **

How many meters do we have in 260cm.

1cm=1.0 x10-2m . Question is asking for meters .

Therefore, Conversion ratio (CR) = [1.0 x 10^{-2}m/1cm] , Given amount is 260cm

Answer = Given amount x CR = 260cm x [1.0 x 10^{-2}m/1cm] =2.6m

__Other Prefixes of SI units__

Decimeters **1dm = 0.1m =1x 10 ^{-1}m **

Milli = Millimeters **1mm=0.001m =1x 10 ^{-3}m**

Micro = Micrometers **1 um = 1x 10^{-6}m ,**

Nano = Nanometers **1nm = 1×10 ^{-9} m **

Pico =Picometers **1pm=1 x 10 ^{-12}m**

Kilo = Kilometers **1km = 1 x 10 ^{3}m , **

Mega = megameters **1 Mm = 1 x 10 ^{6}m ,**

Giga = gigameters **1 Gm = 1 x 10 ^{9}m,**

Tera = terameters **1 Tm = 1 x 10 ^{12}m**

**AREA**

**1cm ^{2} = ? meters square**

**(1cm) ^{2} = (1 x10^{-2} m)^{2 } = 1 x10^{-4} m^{2 } **

**1cm ^{2} = 1 x10^{-4} **

**m**

^{2 }**VOLUME**

**1cm ^{3} = ? meters cube**

**(1cm) ^{3} = (1 x10^{-2} m)^{3 } = 1 x10^{-6} m^{3 } **

** 1cm ^{3} = 1 x10^{-6} m^{3 } **

MASS

SI unit for mass is kilogram.

Example convert 1.0kg into grams

**1000g=1kg** 1.0 x 10^{3} g = 1.0 Kg

Example convert 1.0 milligrams into kilograms.

**Pressure Units**

Pascal (Pa)=N/m^{2}

**760mmHg=760 torrs = 1atm**

How much is 540mmg in atm pressure.

ANSWER:

1atm = 760mmHg ; we are given amount in mmHg and converting into atm units.

Therefore, Conversion Ratio = [1 atm/760 mmHg] .

The given amount is 540mmHg.

Therefore Answer = CR x Given amount = 540mmHg x 1atm/760mmHg = 0.711 atm

**TIME SI UNITS**

**60s =1minute**

**60min =1hour**

How many seconds are there in 365 minutes.

1 minute = 60 s . question is asking for seconds , therefore CR = [60s/1minute] , Given amount is 365 Minutes

Answer = given amount x CR

=365 minutes x [60s/1 minute]

**MULTIPLE UNIT CONVERSIONS**

EXAMPLE: Convert 1g/ml to kilograms/m^{3}

ANSWER:

- You can convert 1.0 grams to kilograms.
- Kilo = 1000 , therefore the Conversion ratio = [1/1000] , amount in kilograms = (1/1000) kilograms = 0.001kg =
**1.0 x 10**^{-3}kg

- Kilo = 1000 , therefore the Conversion ratio = [1/1000] , amount in kilograms = (1/1000) kilograms = 0.001kg =
- Convert 1ml to cubic meters.
- 1 ml = 1 cm
^{3}, c=centi = 1 /100 , therefore, cubic meters = (1/100)^{3}= (0.01)^{3}= (1.0 x 10^{-2})^{3}=**1.0 x 10**^{-6}m^{3}

- 1 ml = 1 cm
- Divide the answer in step 1 by the answer in step 2 .

**(1 x 10**^{-3}kg) / (1 x 10^{-6}m^{3}) = 1.0 x 10^{3}kg/m^{3}

EXAMPLE

Convert the pressure unit of 1.0 psi into N/m^{2}

ANSWER

psi refers to pounds per square inch.

Convert 1.0 pound (lb) into weight in Newtons, then convert 1.0 inches ^{2} into meters^{2} , then divide the value in Newtons by the value in meters^{2} .

1.0lb = 0.454 kg , therefore we convert into weight by multiplying by 9.8m/s^{2} . This gives 0.454 kg x 9.8m/s^{2} = 4.4492 kgm/s^{2} =4.4492 N

1inch = 0.0254 meters , therefore, (1inche)^{2} = (0.0254 meters )^{2} = 6.4516 x 10^{-4} m^{2}

Dividing ; gives 4.4492 N / (6.4516 x 10^{-4} m^{2} ) = 6.9 x 10^{3} N/m^{2}

EXAMPLE :

Convert 23 m/s to miles/hour

ANSWER:

- Convert 23meters into miles
- Convert 1 second to hours
- Divide answer 1 by answer 2.