Thermodynamics of a rubber band

In summary, the conversation discusses a proof that the internal energy of a stretched rubber band is a function of temperature, and that adiabatic stretching of the band results in an increase in temperature. It also mentions that the band will contract if warmed while kept under constant tension. The conversation uses the fundamental equation for a thermoelastic system and the first law of thermodynamics to come to these conclusions.
  • #1
XCBRA
18
0

Homework Statement


For a stretched rubber band, it is observed experimentally that the tension f is proportional tot he temperature T if the length L is held constant. Prove that:

(a) the internal Energy U is a function of temperature;

(b) adiabatic stretching of the band results in an increase in temperature;

(c) the band will contract if warmed while kept under constant tension.

Homework Equations


The Attempt at a Solution



Start with the fundamental equation for a thermoelastic system

du = T ds - f dl.

Then I am stuck as to how to continue from here.

I have tried tp then take the total differential of U:

[tex] du = \frac{\partial U}{\partial S}_Lds +\frac{\partial U}{\partial L}_SdL [/tex]

but that doesn't seem to help. I think I need to use a maxwell relation but I unable to figure out a suitable relationship to do the firs part. Any help will be greatly appreciated.
 
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  • #2
XCBRA said:

Homework Statement


For a stretched rubber band, it is observed experimentally that the tension f is proportional tot he temperature T if the length L is held constant. Prove that:

(a) the internal Energy U is a function of temperature;

(b) adiabatic stretching of the band results in an increase in temperature;

(c) the band will contract if warmed while kept under constant tension.

Homework Equations





The Attempt at a Solution



Start with the fundamental equation for a thermoelastic system

du = T ds - f dl.

Then I am stuck as to how to continue from here.

I have tried tp then take the total differential of U:

[tex] du = \frac{\partial U}{\partial S}_Lds +\frac{\partial U}{\partial L}_SdL [/tex]

but that doesn't seem to help. I think I need to use a maxwell relation but I unable to figure out a suitable relationship to do the firs part. Any help will be greatly appreciated.

(a)
Start with your equation du = T ds - f dl.
What is dl here? Therefore, what is dW?
OK, so then can you rewrite the first law in terms of U and Q, where dQ = Cl*dt?

(b)
First law again! dQ = 0, so how is U affected when W is added to the system?
And what did part (a) say?

(c)
Go back to you 1st equation, now df = 0. How is W, and therefore l, affected?
 

1. What is the thermodynamic behavior of a rubber band?

The thermodynamic behavior of a rubber band is characterized by its ability to stretch and contract in response to changes in temperature. As the temperature increases, the rubber band expands due to increased molecular motion, and as the temperature decreases, the rubber band contracts due to decreased molecular motion.

2. What factors affect the thermodynamics of a rubber band?

The thermodynamics of a rubber band are affected by several factors, including temperature, molecular structure of the rubber, and external forces such as stretching or compressing the rubber band.

3. How does the thermodynamics of a rubber band relate to its elasticity?

The thermodynamics of a rubber band play a crucial role in its elasticity. As the temperature increases, the rubber band becomes more elastic, meaning it can stretch further without breaking. As the temperature decreases, the rubber band becomes less elastic and more brittle.

4. Can the thermodynamics of a rubber band be used to design new materials?

Yes, the thermodynamics of a rubber band can be used to design new materials with specific properties. By understanding how temperature and molecular structure affect the behavior of a rubber band, scientists can develop materials with desired levels of elasticity and flexibility for various applications.

5. Is the thermodynamics of a rubber band a reversible process?

Yes, the thermodynamics of a rubber band is a reversible process. This means that when the temperature changes, the rubber band will expand or contract accordingly, but when the temperature returns to its original state, the rubber band will also return to its original size and shape.

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