Equation of state of a rubber band

[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

Homework Equations

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.

 

[Bystander]

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

 

[Robsta]

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

 

[Robsta]

Alternatively,
du = C_ldT - f dl.

 

[Bystander]

And if L is held constant?

 

[Robsta]

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

 

[Robsta]

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)

 

[Bystander]

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."

 

[Robsta]

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.

 

[Bystander]

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

 

[Robsta]

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

 

[Bystander]

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.

 

[Chestermiller]

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

 

[Chestermiller]

Alternatively,
du = C_ldT - 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