# Equation of state of a rubber band

• Robsta
In summary, for a stretched rubber band with constant length, the tension f is directly proportional to the temperature T. This relationship can be used to show that the internal energy U is only a function of temperature, and that adiabatic stretching of the band leads to an increase in temperature. Additionally, it can be shown that the band will contract when warmed under constant tension. The equation for the internal energy is analogous to that for an ideal gas, and can be written as du = T ds - f dl. By taking the partial derivatives of u with respect to L at constant s, T at constant L, and f with respect to T at constant L, it can be shown that U is indeed only a function of temperature.
Robsta

## 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. Show that:

a) The internal energy U is a function of temperature only
b) Adiabatic stretching of the band results in an increase in temperature (solved)
c) The band will contract if warmed while kept under constant tension

f = kT
(df/dt)L = k

## The Attempt at a Solution

I think I need an equation of state for the band for part (a)

For part (b), adiabatic stretching means stretching with no heat going in or out of the band. To stretch it, the tension must be increased. f = kT:
f1/T1 = f2/T2
T2/T1 = f2/f1 If the band is stretched, f2/f1 is >1 so T2/T1 >1 therefore T2>T1. The temperature has increased.

For part c), I can write the total differential, then let df = 0 but that doesn't give much. Intuitively this makes sense, I just need to formalise it.

Can you write an equation for the internal energy that is analogous to that for an ideal gas?

yes,
du = T ds - f dl.
But that hasn't helped me much.

Alternatively,
du = CldT - f dl.

And if L is held constant?

Do you know how to do this? Because at the moment it looks like internal energy is firmly a function of T and L, not T alone

yes, if L is held constant, U is a function of T. But the question is asking me to show U is a function of just T (presumably with variable L)

Robsta said:
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.
"For an ideal gas it is observed experimentally that the _____ __ is proportional to the temperature T if the ______ __ is held constant."

Pressure is proportional to the temperature if the volume is held constant. So you're saying the U is only a function of T when L is constant and not otherwise? This makes sense, I just thought the question was getting at the fact that if you stretch the band it will heat up too.

And if you squeeze a balloon filled with an ideal gas, does its temperature go up?

No (if it's an ideal gas). But why should I treat the band as ideal?

Robsta said:
treat the band as ideal?
"The thermodynamics of the ideal rubber band" is a popular (not universally used) alternative/generalization of the work term in first law.

Robsta said:
yes,
du = T ds - f dl.
But that hasn't helped me much.
In this equation, what is the partial derivative of u with respect to L at constant s? What is the partial derivative of u with respect to T at constant L? What is the partial derivative of f with respect to T at constant L (in terms of partial derivatives of u)?

Chet

Robsta said:
Alternatively,
du = CldT - f dl.
This equation is incorrect. You need to start with du = T ds - f dl, and then show that:

##du=C_ldT+\left[T\left(\frac{\partial f}{\partial T}\right)_l-f\right]dl##

This is very similar to deriving the general equation for du for a gas:

##du=C_vdT+\left[T\left(\frac{\partial P}{\partial T}\right)_v-P\right]dv##

Chet

## 1. What is the equation of state of a rubber band?

The equation of state of a rubber band is a mathematical relationship that describes the behavior of the rubber band when it is stretched or compressed. It relates the changes in the physical properties of the rubber band, such as length, volume, and pressure, to the applied forces.

## 2. How is the equation of state of a rubber band derived?

The equation of state of a rubber band is derived from the principles of elasticity and thermodynamics. It takes into account the material properties of the rubber band, such as its elasticity, and the external forces applied to it.

## 3. What factors affect the equation of state of a rubber band?

The equation of state of a rubber band is affected by various factors such as temperature, composition of the rubber material, and the type and magnitude of the forces applied to the rubber band. These factors can alter the elasticity and other physical properties of the rubber band, thus changing its equation of state.

## 4. Can the equation of state of a rubber band be used to predict its behavior?

Yes, the equation of state of a rubber band can be used to predict its behavior under different conditions. By plugging in the relevant variables, such as force and temperature, into the equation, we can determine the resulting changes in the physical properties of the rubber band, such as its length or volume.

## 5. How is the equation of state of a rubber band useful?

The equation of state of a rubber band is useful in various fields, such as material science and engineering. It helps in understanding the behavior of rubber materials and designing products that use rubber components. It also has applications in the study of polymer physics and thermodynamics.

• Introductory Physics Homework Help
Replies
2
Views
945
• Introductory Physics Homework Help
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
5
Views
3K
• Introductory Physics Homework Help
Replies
8
Views
896
• Introductory Physics Homework Help
Replies
4
Views
3K
• Introductory Physics Homework Help
Replies
5
Views
5K
Replies
3
Views
5K
• Mechanics
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
20K
• Thermodynamics
Replies
2
Views
2K