# How to Calculate Cubic Expansivity with Examples

## What is Cubic Expansivity

Definition: Cubic expansivity or otherwise volume expansivity, is the increase in the volume of a material per unit volume per degree rise in temperature. Cubic expansivity is denoted by γ. It is the ratio of the change in volume to the product of the original volume by temperature rise. Now, let us have a look at how to calculate cubic expansivity with examples:

## Understanding Cubic Expansivity

Cubic expansivity, often referred to as the coefficient of volume expansion, is a fundamental concept in physics that characterizes how the volume of a substance changes with variations in temperature. It specifically applies to isotropic materials like liquids and solids. When a substance undergoes a temperature change, it generally expands when heated and contracts when cooled.

The cubic expansivity is denoted by the symbol β (beta) and is defined as the fractional change in volume (ΔV/V) per unit change in temperature (ΔT). Mathematically, it is expressed as β = (1/V) * (ΔV/ΔT).

The concept is particularly relevant in understanding thermal expansion phenomena, such as the expansion of substances in response to increased temperature. Different materials have different cubic expansivity values, which influence their thermal behaviour. Solids like metals often have relatively low values of β, making them less prone to drastic volume changes with temperature fluctuations, while liquids like water have higher values of β, resulting in more significant volume changes when heated or cooled.

In practical applications, knowing the cubic expansivity of a material is crucial for designing systems and structures that can withstand temperature variations without detrimental effects. To understand how to calculate cubic expansivity, you will need to know its formula:

### Cubic Expansivity Formula

The cubic expansivity formula is:

Cubic expansivity (γ) = the change in volume (∆V) / ( original volume [V1] x rise in temperature [θ] )

and can be written mathematically as:

γ = ∆V / V1∆θ

where ∆V = V2 – V1

Now,

γ = (V2 – V1) / V1 ( θ2 – θ1 )

and

γ = cubic expansivity

V1 = Original or Initial volume

V2 = Final volume

θ1 = Initial temperature

θ2 = Final temperature

from the above equation, we can see that

γ V1∆θ = V2 – V1 = Change in volume

To make V2 subject of the formula, we now say

V2 = γ V1∆θ + V1

This is also equal to

V2 = V1 (γ∆θ + 1)

Another formula for cubic expansivity is

γ = 3α

and if α = β / 2,

this implies that

γ = 3 x ( β / 2 )

Therefore, we can also use the formula below to calculate cubic expansivity

γ = 3β / 2

### What is apparent Cubic Expansivity

Definition: Apparent cubic expansivity is the ratio of the mass of liquid ejected to the remaining mass when the temperature increases by 10c.

The formula for calculating apparent cubic expansivity is

Apparent Cubic expansivity, γa = ( Mass of the ejected liquid / [mass of the remaining liquid x rise in temperature] )

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## Solved Problems on How to Calculate Cubic Expansivity

Here are a few examples of cubic expansivity questions to make you understand how to calculate cubic expansivity with ease.

### Example 1

What is the cubical expansivity of brass at 600c, if the density at 00c is 15.2g/cm3.

Solution

Data:

We can easily extract our data from the above question

We know that the linear expansivity of brass (αb) = 1.9 x 10-5 K-1

and the formula for cubic or volume expansivity is γ = 3α

Thus γ = 3 x αb

Therefore, we can write

γ = 3 x 1.9 x 10-5 K-1 = 5.7 x 10-5 K-1

### Example 2

A cube with sides 100 centimeters at 00c is heated to 1000c. If the sides becomes 101 centimeter long, find the cubic expansivity of it is material.

Solution

Data:

γ = (V2 – V1) / V1 ( θ2 – θ1 )

and

γ = cubic expansivity = ?

L1 = Original or Initial length = 100 cm

L2 = Final length = 101 cm

θ1 = 00c

θ2 = 1000c

looking at

V = L x b x h = L x L x L

From L1 = 100 cm

This shows that

V1 = 100 x 100 x 100 = 1,000,000 cm3 = 106 cm3

V2 = 101 x 101 x 101 = 1,030,301 cm3 = 106 cm3

Now we apply

γ = (1,030,301 cm3 – 1,000,000 cm3) / 1,000,000 cm3 ( 1000c – 00c )

This is equal to

γ = 30,301 cm3 / 107 cm3 0c

γ = 0.0030301 k-1

### Example 3

The linear expansivity of the material of a cube is 12 x 10-6 k-1. If the length of each side of the cube is 10 centimeters, find the area of one face of the cube and the volume of the cube when it is temperature is raised by 30 k

Solution:

Data

Linear expansivity of the material, α = 12 x 10-6 k-1

This implies that the cubic expansivity of the material is γ = 3α

Which shows that

γ = 3 x 12 x 10-6 k-1 = 36 x 10-6 k-1

and α = β / 2 which implies that β = 2α

Thus

β = 2 x 12 x 10-6 k-1 = 24 x 10-6 k-1

Length of each side of the cube = 10 cm

and area, A = L x b = 10 x 10 = 100 cm2

Volume, V = L x b x h = 10 x 10 x 10 = 1000 cm3

Therefore,

Initial area, A`1 = 100 cm2

Initial volume is V2 = 1000 cm3

Now, we apply the formula below to find the final area

A`2 = A`1 ( 1 + βθ)

which is equal to

A`2 = 100 ( 1 + 24 x 10-6 x 30) = 100 cm2

To calculate the final volume V2, we now use the formula

V2 = V1 (γ∆θ + 1)

By substituting our data into the above equation, we now have

V2 = 100 (36 x 10-6 x 30 + 1) = 1001 cm3

### Example 4

A solid metal cube of side 20 cm is heated from 100c to 1000c. if the linear expansivity of the metal is 1.8 x 10-4 k-1. Calculate the increase in the volume of the metal cube.

Solution:

Data:

Looking at this formula

γ V1∆θ = V2 – V1 = Change in volume = increase in volume = ∆V

we now have

∆V = γ V1∆θ

and ∆θ = θ2 – θ1

From the question, our data stands as

Initial temperature = θ1 = 100c

Final temperature = θ2 = 100c

Linear expansivity = α = 1.8 x 10-4 k-1

Cubic or volume expansivity = γ = 3α = 3 x 1.8 x 10-4 k-1 = 5.4 x 10-4 k-1

Also Volume, V = l x b x h = 10 x 10 x 10 = 1000 cm3 (where l = b = h = 10 cm)

Thus V1 = 1000 cm3

Now, we calculate the increase in volume as

∆V = γ V12 – θ1)

∆V = 5.4 x 10-4 x 1000 (100 – 10) = 48.6 cm3

### Example 5

A blacksmith heated a metal whose cubic expansivity is 6.3 x 10-6 k-1. What is the area expansivity of the metal

Solution

Data

From the above question γ = 6.3 x 10-6 k-1

β = ? and we know that β = 2α

But we don’t have alpha from our question

Therefore we need to first find α

And the relationship between linear expansivity α and cubic (volume) expansivity γ is

γ = 3α

we make α subject of the formula

α = γ / 3 = 6.3 x 10-6 / 3

Hence,

α = 2.1 x 10-6

β = 2α = 2 x 2.1 x 10-6

Therefore, the area expansivity of the metal is

β = 4.2 x 10-6 k-1

### Example 6

A metal of volume 40 centimeter cube is heated from 300c to 900c. What is the increase in volume of the metal.[Linear expansivity (α) = 2.0 x 10-3 k-2 ]

Solution:

Data:

By extracting our data, we have

The initial or original volume, V1 = 40 cm3

Initial temperature = θ1 = 300c

Final temperature = θ2 = 900c

Change in volume, ∆V = ?

∆V = γ V1∆θ

and ∆θ = θ2 – θ1 = (900 – 300) = 600

where γ = 3α = 3 x 2.0 x 10-3 = 0.06

Now we can substitute our data into the main equation

∆V = 0.06 x 40 x 60 = 14.4 cm3

### Example 7

The coefficient of cubical expansion of a cube occupying a volume of 16cm3 is 1.8 x 10-6 k-1. At the same temperature and condition, another metal of volume 8cm3 is used. Find the coefficient of the cubical expansion of the metal.

Solution:

Data:

γ1 = 1.8 x 10-6 k-1

γ2 = ?

V1 = 16cm3

V2 = 8cm3

Since the formula for an increase in volume is

∆V = γ V1∆θ

we can use the relation

γ1V1 = γ2V2

By making γ2 subject of the formula, we find

γ2 = γ1V1 / V2

we now input our values into the above equation

γ2 = (1.8 x 10-6 k-1 x 16cm3 ) / 8cm3

This will give us

γ2 = 0.0000288 / 8

γ2 = 0.0000036 = 3.6 x 10-6 k-1

### Example 8

A density bottle full of liquid is heated from 00c to 1000c during which 6.8g of a liquid is expelled, leaving 400g of it in the bottle. calculate the apparent cubic expansivity of the liquid.

Solution:

Data:

Mass of the ejected liquid, M1 = 6.8g

Also, the mass of the remaining liquid, M2 = 400g

rise in temperature, ∆θ = 100 – 0 = 1000c

Apparent Cubic expansivity, γa = ?

Now we apply the formula below

Apparent Cubic expansivity, γa = ( Mass of the ejected liquid / [mass of the remaining liquid x rise in temperature] )

γa = M1 / ( M2 ∆θ )

after substituting our values into the above equation, we now have

γa = 6.8 / ( 400 x 100 ) = 1.7 x 10-4 k-1

Therefore the apparent cubic expansivity (γa) is 1.7 x 10-4 k-1

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