Cubical Expansion

    VOLUMETRIC EXPANSION
    Increase in volume of a body on heating is referred to as
    VOLUMETRIC EXPANSION ” or “CUBICAL EXPANSION“.
    Consider a metallic body of volume = V1 . Let it’s temperature is raised by T. then experimentally “Increase in volume (V) is directly proportional to its initial volume (V1) and rise in temperature (T).
     V1———(1)
     T———(2)
    combining (1) and (2)
     V1 T
    OR


    Where is constant known as “coefficient of volumetric expansion”
    EXPRESSION FOR FINAL VOLUME
    We know that
    Final volume = Initial volume + increase in volume
    V2 = V1 + V
    Putting the value of V
    V2 = V1 + V1T
    V2 = V1( 1+ T)
    COEFFICIENT OF VOLUMETRIC EXPANSION
    Coefficient of volumetric expansion () is defined as:
    “increase in volume per unit original volume per Kelvin rise in temperature is called coefficient of volumetric expansion.”
    Unit:
    .
    SHOW THAT: 
    .
    Consider a metallic cube of each side of length of “L”.

    Before heating at T1 KVOLUMETRIC EXPANSION……………………
    After heating at T2 KVOLUMETRIC EXPANSION…………………..
    Volume of cube at T1 K :
    V1 = L3 ———- 1
     
    [V = L.B.H]

    If it is heated to T2 K, then the length of each side will become (L + 
    D L) and its volume will become 
    V2 = (L + D L) 3
    since V2= V1 + D V 
    comparing above two equations
    V1 + D V = (L + D L)3
    as [ D L = a L D T ] 

    V1 + D V = (L + a L D T)3 
    V1 + D V = L(1 + a D T)3
    Using formula [(a+b)3 = a3 + b3 + 3a2b + 3ab2]
    V1 + D V = L3 {1 + a 3D T+ 3(1)( a D T) + 3(1)( a 2 D T2)}
    V1 + D V = L3 {1 + a 3D T+ 3a D T + 3a 2 D T2}
    Since “a
      is a fractional value of range 10-5 or 10-6, therefore we
    can neglect all the terms containing high powers of 
    a
      thus.
    V1 + D V = L(1 + 3a D T)Putting the value of DV
    V1 + b V1D T = V1 (1 + 3a D T) Since 
    [D V = V1b D T 
    and L3 
    = V1]


    1 + b D T = 1 + 3a D T



    This expression shows that the coefficient of the cubical expansion is three times of the coefficient of the linear expansion.