Thermal Physics question regarding the thermal expansion coefficients

In summary, the question is about proving that for a solid, the linear thermal expansion coefficients in x, y, and z directions add together to give the thermal expansion coefficient. The equation given is B=ax+ay+az, where B is the thermal expansion coefficient and a is the linear thermal expansion coefficient. The question also asks if the equation should be B=ax*ay*az instead, and if there is a way to use calculus to get the volume from the lengths of an object. The attempt at a solution involves defining the volume as Lx+Ly+Lz and finding the relationship between change in volume and change in temperature. Ultimately, you need to work out what an infinitesimal change in dx, dy, and dz does
  • #1
Stolbik
2
0

Homework Statement




I am working on this ahead of my fall class and don't actually want the answer...
just pointers to help me understand something.. Thanks guys! :)
I am really rusty with my general physics and calculus knowledge =(

The original question asks me to prove that, for a solid, the linear thermal expansion coefficients (in x, y, z directions) add together to give the thermal expansion coefficient as such:

B=ax+ay+az

where B is the thermal expansion coefficient
B=(deltaV/V)/deltaT
deltaV= change in volume
V=volume
deltaT=change in temperature in Kelvin

and a is the linear thermal expansion coefficient
a=(deltaL/L)/deltaT
deltaL= change in length
deltaL=length

So here are my questions:

Shouldn't the equation be B=ax*ay*az instead? for a solid like a cube you have to multiply the lengths to get the volume... Why isn't it the same here?

Also is there a way to get the volume from the lengths of an object with calculus? I don't remember =( Just remember you can get the area under a curve from doing the integral or the volume of an object made by a curve somehow too...

Homework Equations



B=ax+ay+az

where B is the thermal expansion coefficient
B=(deltaV/V)/deltaT
deltaV= change in volume
V=volume
deltaT=change in temperature in Kelvin

and a is the linear thermal expansion coefficient
a=(deltaL/L)/deltaT
deltaL= change in length
deltaL=length

The Attempt at a Solution



uh.. well my attempt so far has been to understand the question. I tried to define the Volume as Lx+Ly+Lz but then got stuck. Please don't give me the answer though! I got 3 months to work this out :)
 
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  • #2
Stolbik said:
Shouldn't the equation be B=ax*ay*az instead? for a solid like a cube you have to multiply the lengths to get the volume... Why isn't it the same here?
You are trying to find the relationship between the change in volume and change in temperature.

You are given the relationship between change in length in the x, y and z directions. From that you can work out what an infinitesimal change of dx, dy and dz does to the change in volume (ie for an infinitesimal change in temperature dT):

V + dV = (x + dx)(y+dy)(z+dz) = (xy + xdy + dxy + dxdy)(z+dz)

Work that out and ignore the second and third order infinitesimal terms.

AM
 

1. What is thermal expansion?

Thermal expansion is the phenomenon where a material expands or contracts when its temperature changes. This is due to the increase or decrease in the average spacing between the atoms or molecules in the material.

2. What is a thermal expansion coefficient?

A thermal expansion coefficient is a measure of how much a material expands or contracts per unit change in temperature. It is typically represented by the symbol α and has units of inverse temperature (1/K or 1/°C).

3. How is the thermal expansion coefficient measured?

The thermal expansion coefficient can be measured by subjecting a material to different temperatures and measuring the corresponding changes in length or volume. The coefficient can also be calculated using the material's thermal properties and its coefficient of linear or volumetric expansion.

4. What factors affect the thermal expansion coefficient of a material?

The thermal expansion coefficient of a material is influenced by various factors such as its chemical composition, crystal structure, and temperature range. Generally, materials with stronger intermolecular forces tend to have lower expansion coefficients, while those with weaker forces have higher coefficients.

5. Why is thermal expansion important in real-world applications?

Thermal expansion is important in many practical applications, such as in building construction, manufacturing of machinery and tools, and in the design of everyday objects like thermometers and thermostats. Understanding the thermal expansion coefficient of materials is crucial in ensuring the stability and functionality of these systems in response to temperature changes.

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