Hello students, we have learnt in previous article about the linear expansion and areal expansion. The coefficient of linear expansion and coefficient of areal expansion are given as below,
Also the relation between coefficient of linear expansion and coefficient of areal expansion is given as,
β= α/2
In this article we are going to discuss the expansion related to change in Volume of substance, i.e. Volume expansion.
Let’s understand concept of Volume expansion in details….!
Increase in Volume of substance under action of heat in known as Volume expansion.
Consider a metal cube of initial Volume V0 at 00 C. Let that cube is being heated to t0 C, so that its final volume becomes Vt. As we know that, here the expansion is along the Volume of cube, then the increase in Volume of cube is given as,
∴ increase in volume = Vt -V0 ……………………….(1)
We can easily observed that when this increase in Volume depends upon following factors,
- Increase in Volume is directly proportional to Volume at 00 C i.e.A0
- Increase in Volume is directly proportional to rise in temperature i.e. ∆t
Then mathematically we can write as,
∴ Vt – V0 α V0
∴Vt – V0 α ∆t
∴Vt – V0 = γ V0.∆t…………… (2)
Where γ is constant of proportionality, here known as coefficient of Volume expansion.
Then from above equation we can write as,
Hence ‘the coefficient of Volume expansion is defined as the increase in Volume of substance per unit Volume at 00 C per unit rise in temperature ∆t’
SI unit of coefficient of Volume expansion is per 0C or /0C
From equation (2) we can get, expression for final Volume of cube as,
∴Vt – V0 = γ V0.∆t
∴Vt = V0 + γ V0.∆t
∴Vt = V0 (1 + γ .∆t) ……………………….(4)
The above formula is for the final Volume of material of cube when its temperature is increased by t0 C.
If V1 and V2 be the initial and final Volumes of cube when heated from at temperature t10 C to t20 C, then we can write,
∴V2 = V1 [1+γ. (t2 – t1)]
The relation between coefficient of linear expansion and coefficient of volume expansion is given as,
γ = α/3
Lets solve some numericals…..!
Ex:1) Find the volume of metal cube of initial volume 45 cm2 heated from 270 C to 2000 C. The coefficient of volume expansion of metal cube is γ = 2.1 × 10-5 /C
Solution:
Here, V1 =45 cm2 = 45 × 10-4 m2, t1 = 270 C ,t2= 2000 C,
∴∆t = t2 – t1 = 1730 C , γ = 2.1× 10-5 /C
We have,
∴V2 = V1 [1+γ. (t2 – t1)]
∴V2 = 45 × 10-4 [1+2.1× 10-5×173]
∴V2 = 45 × 10-4 [1+363.3× 10-5]
∴V2 = 45 × 10-4 [1+0.00363.3]
∴V2 = 45× 10-4 [1.003633]
∴V2 = 45.163 m2