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 (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.
Solution:
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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