## Introduction

Cubic expansivity otherwise known as volume expansivity, is the increase in volume per unit volume per degree rise in temperature. Here are simple steps we need to take in order to understand how to derive the formula for increase in volume from cubic expansivity.

We use a Greek alphabet gamma (Î³) to denote cubic expansivity.

We can write cubic (volume) expansivity as Î³ **=** ( V_{2 }â€“ V_{1}) **/ **(V_{1}âˆ†Î¸)â€¦â€¦.(Equation 1)

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

Addtionally, Î³ = 3Î± ( Because Î± = Î² / 2 )â€¦â€¦â€¦(Equation 2)

From Equation 1, we can further break it into

( V_{2 }â€“ V_{1}) = Î³ (V_{1}âˆ†Î¸) â€¦â€¦.. (Equation 3)

Where âˆ†Î¸ = change in temperature = ( Î¸_{2} â€“ Î¸_{1} )

Therefore, ( if the change in volume V_{2 }â€“ V_{1 }= âˆ†V )

This implies that Equation 3 will become

âˆ†V = Î³ (V_{1}âˆ†Î¸)â€¦â€¦â€¦ (Equation 4)

Now going back to Equation 3 which is ( V_{2 }â€“ V_{1}) = Î³ (V_{1}âˆ†Î¸)

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

V_{2 } = V_{1}(1 + Î³âˆ†Î¸)â€¦â€¦â€¦â€¦..(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 Î¸_{1} = 10^{0}C

Final temperature Î¸_{2} = 60^{0}C

Since âˆ†Î¸ = change in temperature = ( Î¸_{2} â€“ Î¸_{1} ) = (60^{0}C â€“ 10^{0}C) = 50^{0}C

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

To calculate the increase in volume, we consider the equation âˆ†V = Î³ (V_{1}âˆ†Î¸)

where âˆ†V = increase in volume

âˆ†V = Î³ (V_{1}âˆ†Î¸) and remember that Î³ = 3Î±

which implies that

âˆ†V = 3Î± (V_{1}âˆ†Î¸)

Now, putting our data into the above equation

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

âˆ†V = 1.8cm^{3}

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