How to Calculate Volume Thermal Expansion Coefficient?

In summary, the problem is asking to prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions. This can be shown by using the equation dV/V = ((dLx*dLy*dLz)/(Lx*Ly*Lz)) and simplifying it to dV= (Lx +dLx)( Ly +dLy)( Lz +dLz) - Lx*Ly*Lz. By neglecting small terms, we can see that dV/V is equal to the sum of the linear expansion coefficients in each direction. This is also simplified by the fact that for an isotropic solid, B = 3A, making the problem easier
  • #1
phrygian
80
0

Homework Statement


Prove that the volume thermal expansion coefficent of a solid is equal to the sum of its linear expansion coefficients in the three directions: B = Ax + Ay + Az


Homework Equations



B = (dV/V)/dT
A = (dL/L)/dT

The Attempt at a Solution



My thought was using dV/V = ((dLx*dLy*dLz)/(Lx*Ly*Lz)) but when you use this it is clear that dL^3/L^3 does not equal 3* dL/L

What am I doing wrong?

Thanks for the help
 
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  • #2
dV= ( Lx +dLx)( Ly +dLy)( Lz +dLz) - Lx*Ly*Lz
Simplify this and proceed.
 
  • #3
Can you explain mathematically how you got there?

Also, the problem statement says at the end (So for an isotropic solid, which expands the same in all directions, B = 3A) if that makes the problem simpler.
 
  • #4
Lx, Ly and Lz are the lengths of the block. When the temperature is raised through 1 degree C, the new lengths will be Lx + Ax, Ly + Ay and Lz+ Az.
New volume will be (Lx + Ax)( Ly + Ay)( Lz+ Az). Original volume is Lx*Ly*Lz.
Sp dV = (Lx + Ax)( Ly + Ay)( Lz+ Az) - Lx*Ly*Lz
Find dV/V. Neglect the terms like Ax*Ay and so on because they are very small quantities.
 
  • #5
rl.bhat said:
dV= ( Lx +dLx)( Ly +dLy)( Lz +dLz) - Lx*Ly*Lz
Simplify this and proceed.

phrygian said:
Can you explain mathematically how you got there?

dV = (volume after expansion) - (initial volume)​

After expansion, the box dimensions have increased by dLx, dLy, and dLz, from their initial lengths Lx, Ly, and Lz.

EDIT:
rl responded faster than I. :smile:
 

What is thermal expansion coefficent?

Thermal expansion coefficent is a measure of how much a material will expand or contract in response to a change in temperature. It is typically represented by the symbol alpha and is expressed in units of length per degree Celsius (or Kelvin).

How is thermal expansion coefficent calculated?

Thermal expansion coefficent is calculated by dividing the change in length of a material by its original length and the change in temperature. This can be expressed as: alpha = (change in length) / (original length * change in temperature).

Why is thermal expansion coefficent important?

Thermal expansion coefficent is important because it can affect the performance and durability of materials in various applications. For example, if a material has a high coefficent of thermal expansion, it may be prone to warping or cracking when exposed to temperature changes.

How does thermal expansion coefficent vary among different materials?

The thermal expansion coefficent varies among different materials due to their atomic and molecular structures. Generally, materials with stronger bonds between atoms tend to have lower coefficents of thermal expansion, while those with weaker bonds have higher coefficents.

Can thermal expansion coefficent be controlled or minimized?

Yes, thermal expansion coefficent can be controlled or minimized through careful material selection and design. For example, using materials with similar coefficents of thermal expansion in a product can help reduce stress and prevent damage due to temperature changes.

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