## Introduction

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.

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

We can write cubic (volume) expansivity as γ **=** ( V_{2 }– V_{1}) **/ **(V_{1}∆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

( V_{2 }– V_{1}) = γ (V_{1}∆θ) …….. (Equation 3)

Where ∆θ = change in temperature = ( θ_{2} – θ_{1} )

Therefore, ( if the change in volume V_{2 }– V_{1 }= ∆V )

This implies that Equation 3 will become

∆V = γ (V_{1}∆θ)……… (Equation 4)

Now going back to Equation 3 which is ( V_{2 }– V_{1}) = γ (V_{1}∆θ)

We can further rearrange the above expression by making V_{2 } subject of the formula

V_{2 } = V_{1}(1 + γ∆θ)…………..(Equation 5)

## Example 1: Calculate Increase in Volume From Cubic Expansivity

A solid metal cube of side 10cm is heated from 10^{0}C to 60^{0}C. If the linear expansivity of the metal is 1.2 x 10^{-5}k^{-1}.

Calculate the increase in its volume.

**Solution:**

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

Length (L) = 10cm and Volume = L x L x L = L^{3 }= (10cm)^{3} = 1000cm^{3}

Initial temperature θ_{1} = 10^{0}C

Final temperature θ_{2} = 60^{0}C

Since ∆θ = change in temperature = ( θ_{2} – θ_{1} ) = (60^{0}C – 10^{0}C) = 50^{0}C

Linear expansivity α = 1.2 x 10^{-5}k^{-1}

To calculate the increase in volume, we consider the equation ∆V = γ (V_{1}∆θ)

where ∆V = increase in volume

∆V = γ (V_{1}∆θ) and remember that γ = 3α

which implies that

∆V = 3α (V_{1}∆θ)

Now, putting our data into the above equation

∆V = 3 x 1.2 x 10^{-5}k^{-1} (1000cm^{3} x 50^{0}C)

∆V = 1.8cm^{3}

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