20.0.0.  AREA EXPANSIVITY:
Area expansivity is the increase in area per unit area of a substance for every one degree Celsius/ kelvin rise in temperature.

UNIT OF AREA EXPANSIVITY:
The unit of area expansivity is per degree Celsius (°C⅃−¹) or per kelvin ( K−¹ )

FORMULAR FOR CALCULATING AREA EXPANSIVITY:
Area expansivity ß = ∆ Area / ( original area* ∆θ  )
Area expansivity ß = change in area / ( original area* temperature change)
Area expansivity ß = increase in area / ( original area * temperature change )
Area expansivity ß = ( final area - initial area ) / ( original area * temperature change)
Area expansivity ß = ( A₂ - A₁  )  /  ( A₁  *  ( θ₂ - θ₁ ) )                                 
                                         ß = ( A₂ - A₁  )  /  ( A₁ *∆θ )
                                         ß * A₁ *∆θ =   A₂ - A₁    
                                         ß * A₁ *∆θ + A₁  =   A₂   ( Collect like terms together )
                                         A₁ ( ß *∆θ + 1 )  =   A₂    ( factorize  A₁ )
                                           ß *∆θ + 1   =   A₂ / A₁
A₁ is the initial or original area of the object at temperature   θ₁
A₂ is the final area or new area at temperature  θ₂ 
∆θ = θ₂ - θ₁. It is called temperature change or change in temperature.
ß Is the area expansivity or superficial expansivity of the material. Its unit is per kelvin.

RELATIONSHIP BETWEEN LINEAR AND AREA EXPANSIVITY:
Area expansivity = 2 * Linear expansivity
                    ß = 2 * ∝

DERIVATION OF THE RELATIONSHIP BETWEEN LINEAR EXPANSIVITY AND AREA  EXPANSIVITY:
Let us consider a sheet of metal of length l and breadth b. Let the initial length be l₁ and the  initial breadth be b₁. Let the final length be l₂ and the final breadth be b₂ after the metal is heated through a temperature change of θ.
Let the initial area of the metal be A₁ = l₁*b₁
Let the final area of the metal be A₂ = l₂*b₂ 

Then,
From the formula of linear expansivity ,   
                      l₂ =  l₁ (∝∆θ + 1 ) 
                      b₂ =  b₁ (∝∆θ + 1 ) 
Recall that Area = l * b.
Therefore,
                       A₁ = l₁*b₁
                       A₂ = l₂*b₂   . 
                       A₂ =l₁ (∝∆θ + 1 ) * b₁ (∝∆θ + 1 )  
                      A₂  = l₁* b₁ (∝∆θ + 1 ) *  (∝∆θ + 1 ) 
                 A₂  = l₁* b₁  ((∝∆θ   *  (∝∆θ + 1 ) + 1 * (∝∆θ + 1 ) )
Let us open the brackets, then we will get,
                 A₂  = l₁* b₁ (∝²(∆θ)² + ∝∆θ  + ∝∆θ + 1 )
                  A₂  = l₁* b₁ (∝²(∆θ)² + 2∝∆θ  +1 )
                   A₂  =  A₁(∝²(∆θ)² + 2∝∆θ  +  1 )     
Recall that ,
                        A₂ =  A₁ ( ß *∆θ + 1 )     
Let us substitute for A₂ in the formula above. Then we will get,
                        A₁ ( ß *∆θ + 1 ) =  A₁(∝²(∆θ)² + 2∝∆θ  +  1 )
                   ß *∆θ + 1 =  ∝²(∆θ)² + 2∝∆θ  +  1    ( A₁ canceled A₁and the brackets are removed )
                        ß *∆θ  =  ∝²(∆θ)² + 2∝∆θ    ( 1 cancelled 1 )
At this point, ∝tends to zero because its value is from 0.0000... Upward. Therefore, ∝² is approximately zero. I.e. ∝² = 0.
Therefore, we will put zero in the place of  ∝²,then we will get,
                       ß *∆θ  =  0 * (∆θ)² + 2∝∆θ 
                  ß *∆θ  =  0 +  2∝∆θ   ( anything multiply by zero is zero )
                  ß *∆θ  =   2∝∆θ    
                  ß   =  2∝    ( ∆θ cancelled ∆θ )
Which shows that,
Area expansivity = 2 times Linear expansivity


APPLICATION OF FORMULA:
EXAMPLES:

1. Calculate the area expansivity of a mental plate whose area at 36°C is 0.25 m² and 1.15m² at 56.2 °C.
2. SOLUTION:    
3. Data given in the question: 
4.  initial temperature 

2. 

VOLUME / CUBIC EXPANSIVITY:

Cubic / volume expansivity is the increase in volume per unit volume of a substance for every one degree Celsius rise in temperature.

FORMULA FOR CALCULATING VOLUME EXPANSIVITY:

Volume expansivity = ∆V / ( original volume * ∆θ )

Volume expansivity = Change in volume / ( original volume * temperature change )

Volume expansivity = increase In Volume  / ( original volume * temperature change )

Volume expansivity = ( final volume - original volume) / ( original volume * ( θ₂ - θ₁ ) )

Volume expansivity =  ( V₂ - V₁  )  /  ( V₁  *  ( θ₂ - θ₁ ) )
Volume expansivity ( ϒ)  =  ( V₂ - V₁  )  /  ( V₁  *   ∆θ )

                                         ϒ  =  ( V₂ - V₁  )  /  ( V₁  *   ∆θ )

                                                    ϒ * V₁ * ∆θ   =   V₂ - V₁         

                                                     ϒ * V₁ * ∆θ + V₁  =  V₂   ( Collect like terms together )

                                                     V₁ (  ϒ * ∆θ + 1 )  =  V₂    ( Factorize V₁ )

                                                     ( ϒ * ∆θ + 1 )  =  V₂ / V₁

V₁ is the initial/ original volume at temperature  θ₁ 

V₂ is the final volume or new volume at temperature θ₂ 
 θ₁ is Initial  temperature  
 θ₂

∆θ =   θ₂ - θ₁   . it is called temperature change or change in temperature.

ϒ is called cubic or volume expansivity of the material. Its unint is per Kelvin

RELATIONSHIP BETWEEN LINEAR EXPANSIVITY AND CUBIC EXPANSIVITY:

Cubic expansivity = 3 Linear expansivity
                             ϒ = 3*∝

DERIVATION OF THE RELATIONSHIP BETWEEN LINEAR EXPANSIVITY AND CUBIC EXPANSIVITY:
Let us consider a sheet of metal of length l , breadth b and height h.Let the initial length be l₁ , the  initial breadth be b₁ and the initial height be h₁. Let the final length be l₂ , the final breadth be b₂ and the final height be h₂, after the metal is heated through a temperature change of θ.
Let the initial volume of the metal be V₁ = l₁*b₁*h₁
Let the final volume of the metal be V₂ = l₂*b₂*h₂

Then,
From the formula of linear expansivity ,   
                      l₂ =  l₁ (∝∆θ + 1 ) 
                      b₂ =  b₁ (∝∆θ + 1 ) 
                      h₂ =  h₁ (∝∆θ + 1 ) 
Recall that Volume = l * b*h
Therefore,
                       V₁ = l₁*b₁*h₁
                       V₂ = l₂*b₂ *h₂  
                       V₂ =l₁ (∝∆θ + 1 ) * b₁ (∝∆θ + 1 ) *  h₁ (∝∆θ + 1 ) 
                      V₂ = l₁* b₁*h₁ (∝∆θ + 1 ) *  (∝∆θ + 1 ) * (∝∆θ + 1 ) 
                 V₂ = l₁* b₁*h₁ ((∝∆θ   *  (∝∆θ + 1 ) + 1 * (∝∆θ + 1 ) ) * (∝∆θ + 1 ) 
Let us open the brackets, then we will get,
                  V₂ = l₁* b₁*h₁(∝²(∆θ)² + ∝∆θ  + ∝∆θ + 1 ) * (∝∆θ + 1 ) 
                  V₂ = l₁* b₁*h₁(∝²(∆θ)² + 2∝∆θ  +1 ) * (∝∆θ + 1 ) 
                  V₂ = l₁* b₁*h₁(∝²(∆θ)² * (∝∆θ + 1 )  +  2∝∆θ * (∝∆θ + 1 ) +1* (∝∆θ + 1 ) )  
                   V₂ = l₁* b₁*h₁(∝³(∆θ)³ +∝²(∆θ)²  +  2∝²(∆θ)² +  2∝∆θ  +∝∆θ + 1 )    
                   V₂ = l₁* b₁*h₁(∝³(∆θ)³ +∝²(∆θ)²  +  2∝²(∆θ)² +  3∝∆θ  + 1 )    
                   V₂ =  V₁(∝³(∆θ)³ + 2∝²(∆θ)² + 3∝∆θ+1 )     
Recall that ,
                       V₂ = V₁ (  ϒ * ∆θ + 1 )        
Let us substitute for V₂ in the formula above. Then we will get,
                        V₁ (  ϒ * ∆θ + 1 ) =  V₁(∝³(∆θ)³ + 2∝²(∆θ)² + 3∝∆θ+1 ) 
                  ϒ * ∆θ + 1 = ∝³(∆θ)³ + 2∝²(∆θ)² + 3∝∆θ+1 (V₁ canceled V₁ and the brackets are removed )
                        ϒ * ∆θ  = ∝³(∆θ)³ + 2∝²(∆θ)² + 3∝∆θ   ( 1 cancelled 1 )
                    
At this point,∝tends to zero because its value is from 0.0000... Upward. Therefore,∝² and ∝³  are approximately zero. I.e. ∝² and ∝³ are = 0.
Therefore, we will put zero in the place of  ∝² and ∝³. Then we will get,
                  ϒ * ∆θ  = 0*(∆θ)³ + 2*0*(∆θ)² + 3∝∆θ
                       ϒ * ∆θ  = 0 + 0 + 3∝∆θ 
                       ϒ * ∆θ  =  3∝∆θ 
                       ϒ   =    3∝  ( ∆θ cancelled. ∆θ )
Therefore, 

Cubic / volume expansivity = 3 Linear expansivity

                       

APPLICATION OF FORMULA IN SOLVING PROBLEMS:

EXAMPLES:
1.  A piece of mass 170kg has its temperature raised from 0°C to 30°C. Calculate its increase in volume, given the density of brass at 0°C at 8.5 x 103 kgm-³ and its cubic expansivity as 57 x 10−⁴K-¹.

SOLUTION:

          Data given  in the question:

          mass = 170kg, initial temperature  θ₁= 0°C, final temperature  θ₂ =  30°C

          density of brass at 0°C = 8.5 x 10³ kgm-³, cubic expansivity =  57 x 10−⁴K-¹.

          ∆θ =  θ₂ - θ₁ = 30° - 0° = 30°

Note:

In this question,  you are to calculate the increase in volume. You must first calculate the volume V₂ at  temperature 30°C. The  volume V₁ at temperature 0°C is not given. But the mass and density of the object at temperature 0°Ç is given. Use these mass and the density of the object to calculate the volume V₁ at temperature 0°C.

Therefore,

                    Density = mass / volume

                    8.5 x 10³ = 170 / Volume

                    Volume =  170 / 8.5 x 10³ 

                          Volume = 0.02m³
                          Volume V₁ = 0.02m³

Now that we have calculated V₁ , we will then calculate V₂ before we will be able to calculate the rise in volume.
Therefore,

Formula:       
                    V₁ (  ϒ * ∆θ + 1 )  =  V₂ 
Substitution:
                     0.02 ( 57 x 10−⁴* 30 + 1 ) = V₂
                1.14e−6 + 0.02 = V₂      ➡   V₂  =  0.02000114m³
Now we will calculate the increase in volume:
                Increase in volume ( ∆V) = V₂ - V₁
                                                        = 0.02000114 - 0.02
                                                        = 0.00000114 m᪘³

2. A solid metal cube of side 10cm is heated from 10°C to 60°C. if the linear expansivity of the metal is 1.2 x 10⁴ K-¹, calculate the increase in its volume.

SOLUTION:
Data given in the question:
Length of metal = 10cm, initial temperature =  10°C, final temperature = 60°C 
inear expansivity of the metal is 1.2 x 10−⁴ K-¹.

Note:
The material is a cube and we are given only one side of the cube. Since a cube has equal sides, the length, breadth and height of the cube are the same. We will use the length to first calculate the initial volume of the cube at 10°C. After that, we will calculate  the volume of the cube at 60°C and thereafter calculate the increase in volume of the cube.

Volume of cube at 10°C , V₁= l*b*h. = 10*10*10 = 1000 cm³,  ∆θ = θ₂ - θ₁ = 60 - 10 = 50°C
                Cubic expansivity = 3 * Linear expansivity
                Cubic expansivity = 3 * 1.2 x 10−⁴ 
                Cubic expansivity  = 0.000036 K−¹

                Formula:           V₂  =  V₁ (  ϒ * ∆θ + 1 )  
                 Substitution:      V₂  = 1000 ( 0.000036  *  50 + 1 )
                                           V₂  = 1000 ( 0.0018 + 1 )   
                                           V₂  = 1000 ( 1.0018)      
                                           V₂  = 10018 cm³                     
                Increase in volume = V₂ - V₁  =  10018 - 1000 = 18 cm³

3. If the cubic expansivity of brass between 270C and 1000C is 5.7 x10-5K-1, what is its linear expansivity?

SOLUTION:
Data given in the question:
cubic expansivity of brass = 0.57 x10−⁴K-¹,  θ₁ = 27°C,  θ₂  = 100°C

Formula:
               cubic expansivity = 3 * Linear expansivity
               0.57 x10−⁴ = 3 * Linear expansivity
               Linear expansivity =  0.57 x10−⁴  / 3
               Linear expansivity  = 0.19 x10−⁴K−¹


4. A solid material of volume 1000cm3 is heated through a temperature difference of 400C. Calculate the increase in the volume of the material if its linear expansivity is 2.0 x10-6K-¹

SOLUTION:
Data given in the question:
Initial volume V₁ = 1000cm³, temperature change = ∆θ = θ₂ - θ₁ = 40°C, V₂ = ?,
linear expansivity =  0.20 x10-⁴K-¹, cubic expansivity = 3 * Linear expansivity
cubic expansivity = 3 * 0.20 x10-⁴K-¹ = 0.6 x10-⁴K-¹ 

Formula:                 V₂  =  V₁ (  ϒ * ∆θ + 1 )  
Substitution:          V₂  =  1000( 0.6 x10-⁴ * 40 + 1 )          
                                 V₂  =  1000( 0.0024 + 1 )         
                                 V₂  =  1000 ( 1.0024 )
                                 V₂  =  1002.4 cm³
                                 Increase in volume = V₂ - V₁  = 1002.4 = 2.4 cm³

EXERCISES:
1.  A piece of mass 170kg has its temperature raised from 00C to 300C. Calculate its increase in volume, given the density of brass at 00C at 8.5 x 103 kgm-3 and its cubic expansivity as 5.7 x 10-5K-1.

2.  A solid metal cube of side 10cm is heated from 10 0C to 60 0C. if the linear expansivity of the metal is 1.2 x 105 K-1, calculate the increase in its volume..

3.  A cube made of metal of linear expansivity is warmed through a temperature of t. If the initial volume of the cube is V, what is the increase in volume of the cube?

4.  Which of the following is the expression for superficial expansivity of a material?

5.