# How Does Temperature Affect the Length of a Rubber Band Under Constant Tension?

• derrickb
In summary, the problem involves finding the fractional change in length (L-L0) of a rubber band model due to an increase in temperature (δT) at constant tension. The equations used are U=cL0T, τ=bT((L-L0)/(L1-L0)), and d/dL(1/T)=d/dU(-τ/T), and the attempt at a solution involves using Maxwell's 4th equation and rewriting the 1st law to show that T dS = Cτ dT if τ is constant. The assumption of an "ideal rubber band" may also be made, allowing for the use of dU = CL dT.
derrickb

## Homework Statement

For the rubber band model, calculate the fractional change in (L-L0) that results from an increase δT in temperature, at constant tension. Express the result in terms of the length and temperature.

## Homework Equations

U=cL0T
τ=bT((L-L0)/(L1-L0)); τ=tension, L1=elastic limit
d/dL(1/T)=d/dU(-τ/T)

## The Attempt at a Solution

I'm sort of at a loss on this one. I've tried subbing in all sorts of equations, but can't seem to make any real progress.

derrickb said:

## Homework Statement

For the rubber band model, calculate the fractional change in (L-L0) that results from an increase δT in temperature, at constant tension. Express the result in terms of the length and temperature.

## Homework Equations

U=cL0T
τ=bT((L-L0)/(L1-L0)); τ=tension, L1=elastic limit
d/dL(1/T)=d/dU(-τ/T)

## The Attempt at a Solution

I'm sort of at a loss on this one. I've tried subbing in all sorts of equations, but can't seem to make any real progress.

EDIT:

By using Maxwell's 4th equation you can show that T dS = Cτ dT if τ is constant.

You can also rewrite the 1st law as dU = T dS + τ dL.

Just thinking - if we can assume an "ideal rubber band" analogously to an ideal gas, such that U is a function of T only, then dU = CL dT similar to dU = CV dT for an ideal gas.

Last edited:

## 1. How does a rubber band stretch and contract?

The stretching and contracting of a rubber band is based on thermodynamics principles. When a rubber band is stretched, the polymer chains within the band are pulled apart, causing an increase in entropy. This increase in entropy is accompanied by an increase in the band's temperature, which allows the polymer chains to move more freely and stretch. When the stretching force is removed, the polymer chains return to their original state, causing the rubber band to contract.

## 2. What is the relationship between temperature and a rubber band's elasticity?

The elasticity of a rubber band is dependent on its temperature. As the temperature of a rubber band increases, its elasticity decreases. This is because the increased temperature causes the polymer chains to move more freely and become less organized, resulting in a decrease in the band's ability to return to its original shape after being stretched.

## 3. How does the length of a rubber band affect its thermodynamic properties?

The length of a rubber band is directly related to its thermodynamic properties. A longer rubber band will have a higher entropy and a lower elasticity compared to a shorter rubber band. This is because the longer rubber band will have more polymer chains that can move and stretch, resulting in a higher increase in entropy and a lower ability to return to its original shape.

## 4. Can a rubber band's thermodynamic properties be affected by external factors?

Yes, a rubber band's thermodynamic properties can be affected by external factors such as temperature, humidity, and exposure to chemicals. These external factors can impact the polymer chains within the rubber band, causing changes in its entropy, elasticity, and other thermodynamic properties.

## 5. How does the thermodynamics of a rubber band relate to its practical uses?

The understanding of the thermodynamics of a rubber band is essential for its practical uses. For example, knowing how temperature affects a rubber band's elasticity can help in choosing the right type of rubber band for a specific application. Furthermore, understanding the thermodynamic properties of a rubber band can also aid in its design and development for various uses, such as in rubber band-powered toys or medical devices.

Replies
3
Views
5K
Replies
2
Views
2K
Replies
4
Views
6K
• Biology and Chemistry Homework Help
Replies
2
Views
326
Replies
1
Views
13K
Replies
1
Views
4K
• Engineering and Comp Sci Homework Help
Replies
8
Views
3K
Replies
6
Views
5K
• Classical Physics
Replies
4
Views
847
• Introductory Physics Homework Help
Replies
5
Views
3K