Adiabatic stretching of a rubber band

In summary, it is observed that for a stretched rubber band, tension is proportional to temperature when length is held constant. Adiabatic stretching of the band results in an increase in temperature, and warming the band while keeping constant tension will cause it to contract. The first law of thermodynamics and the equations of state for the rubber are used to prove these relationships.
  • #1
Toby_phys
26
0

Homework Statement


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

(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 first law:
$$
dU=Tds+fdL=C_L dT
$$
$$
f=\left (\frac{\partial f}{\partial T}\right )_L T
$$
$$
\left (\frac{\partial L}{\partial f}\right )_T>0
$$

The Attempt at a Solution



(b)[/B]

For an adiabatic process, entropy doesn't increase and so:
$$
dU=fdL=C_LdT
$$

The force is always positive and so temperature is positively increased by length.

This feels too simple so i doubt I am correct. I have no idea for part (c).
 
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  • #2
How do you know that dU=CdT?
 
  • #3
There was a part A that was to show this is the case
$$dU=C_vdT+\left[f-T\left(\frac{\partial f}{\partial T}\right)_L\right]dL$$

The second term drops out
 
  • #4
Toby_phys said:

Homework Statement


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

(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 first law:
$$
dU=Tds+fdL=C_L dT
$$
$$
f=\left (\frac{\partial f}{\partial T}\right )_L T
$$
$$
\left (\frac{\partial L}{\partial f}\right )_T>0
$$

The Attempt at a Solution



(b)[/B]

For an adiabatic process, entropy doesn't increase and so:
$$
dU=fdL=C_LdT
$$

The force is always positive and so temperature is positively increased by length.

This feels too simple so i doubt I am correct. I have no idea for part (c).
In part (a) it was shown that f = Tg(L), where g is an increasing function of L. This is the equation of state of the rubber. So if T increases at constant f, what happens to L?
 

Related to Adiabatic stretching of a rubber band

1. What is adiabatic stretching of a rubber band?

Adiabatic stretching of a rubber band is the process of stretching a rubber band without any heat exchange with the surrounding environment. This means that the temperature of the rubber band remains constant throughout the stretching process.

2. How does adiabatic stretching affect the rubber band?

When a rubber band is stretched adiabatically, it experiences an increase in its internal energy. This results in an increase in the length and stiffness of the rubber band.

3. What factors affect the adiabatic stretching of a rubber band?

The adiabatic stretching of a rubber band is affected by the material properties of the rubber band, such as its elasticity and thermal conductivity, as well as the initial and final lengths of the rubber band.

4. What is the significance of adiabatic stretching of a rubber band?

Adiabatic stretching of a rubber band is important in studying the behavior of rubber materials and their potential applications. It is also used in various industrial processes, such as the production of rubber products.

5. Can adiabatic stretching cause a rubber band to break?

Yes, adiabatic stretching can cause a rubber band to break if it is stretched beyond its elastic limit. This is because the increase in internal energy can weaken the rubber band, making it more susceptible to breaking under tension.

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