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How to Derive the Formula For Increase in Volume


In this post, i will guide you on how to derive the formula for increase in volume from cubic expansivity

Cubic expansivity otherwise known as volume expansivity, is the increase in volume per unit volume per degree rise in temperature.

How to Derive the Formula For Increase in Volume
How to Derive the Formula For Increase in Volume

We use a Greek alphabet gamma (γ) to denote cubic expansivity.

We can write cubic (volume) expansivity as γ = ( V2 – V1) / (V1∆θ)…….(Equation 1)

Also, from the above definition, Cubic = Change in Volume / Original Volume x Temperature

Addtionally, γ = 3α ( Because α = β / 2 )………(Equation 2)

From Equation 1, we can further break it into

( V2 – V1) = γ (V1∆θ) …….. (Equation 3)

Where ∆θ = change in temperature = ( θ2 – θ1 )

Therefore, ( if the change in volume V2 – V1 = ∆V )

This implies that Equation 3 will become

∆V = γ (V1∆θ)……… (Equation 4)

Now going back to Equation 3 which is ( V2 – V1) = γ (V1∆θ)

We can further rearrange the above expression by making V2 subject of the formula

V2 = V1(1 + γ∆θ)…………..(Equation 5)

Example 1: Calculate Increase in Volume From Cubic Expansivity

A solid metal cube of side 10cm is heated from 100C to 600C. If the linear expansivity of the metal is 1.2 x 10-5k-1.

Calculate the increase in its volume.


Data: We need to understand the question and extract our data

Length (L) = 10cm and Volume = L x L x L = L3 = (10cm)3 = 1000cm3

Initial temperature θ1 = 100C

Final temperature θ2 = 600C

Since ∆θ = change in temperature = ( θ2 – θ1 ) = (600C – 100C) = 500C

Linear expansivity α = 1.2 x 10-5k-1

To calculate the increase in volume, we consider the equation ∆V = γ (V1∆θ)

where ∆V = increase in volume

∆V = γ (V1∆θ) and remember that γ = 3α

which implies that

∆V = 3α (V1∆θ)

Now, putting our data into the above equation

∆V = 3 x 1.2 x 10-5k-1 (1000cm3 x 500C)

∆V = 1.8cm3

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