How Do You Derive the Energy Equation for a Rubber Band?

In summary: I didn't make this up, this is correct. The sign of \mathcal{F} is negative, so it makes dT smaller than dU.
  • #1
decerto
87
2

Homework Statement



The equation of state for a rubber band with temperature T is [itex]\mathcal{F}=aT\left[\frac{L}{L_0}-\left(\frac{L_0}{L}\right)^2\right][/itex]

Where [itex]\mathcal{F}[/itex] is the tension, L is the stretched length and L_0 is the unstretched length

a) Write the Central Equation for the rubber band

b) Derive the energy equation for the rubber band [itex]\left(\frac{\partial U}{\partial L}\right)_T[/itex]

c) Show that U is a function of T only

Homework Equations



[itex] dU=dQ+dW[/itex]

[itex] dU=\left(\frac{\partial U}{\partial T}\right)_LdT +\left(\frac{\partial U}{\partial L}\right)_TdL [/itex]

The Attempt at a Solution



a) For the central equation some variant of [itex] dU=TdS+\mathcal{F}dL [/itex] I pressume?

b) Comparing the two relevant equations [itex]\left(\frac{\partial U}{\partial L}\right)_T=\mathcal{F} [/itex] ?

c)No real idea how to show this
 
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  • #2
decerto said:
a) For the central equation some variant of [itex] dU=TdS+\mathcal{F}dL [/itex] I pressume?

OK

b) Comparing the two relevant equations [itex]\left(\frac{\partial U}{\partial L}\right)_T=\mathcal{F} [/itex] ?

This is incorrect. Note that from [itex] dU=TdS+\mathcal{F}dL [/itex] you can get [itex]\left(\frac{\partial U}{\partial L}\right)_S=\mathcal{F} [/itex]. But you need to get an expression for [itex]\left(\frac{\partial U}{\partial L}\right)_T [/itex].

One approach is to rearrange your central equation for dS and then use one of your relevant equations to substitute for dU. That should express dS in terms of dT and dL. Then try to see what to do.

c)No real idea how to show this

Once you get the answer for (b) this will be easy.
 
  • #3
With [itex]dS=\frac{1}{T}dU-\frac{F}{T}dL[/itex] I'm slightly confused on how to incorporate dT [itex]dU=\left(\frac{\partial U}{\partial T}\right)_LdT +\left(\frac{\partial U}{\partial L}\right)_TdL[/itex] is wrong yes? I think I just made that up.
 
  • #4
Ok how about this.

[itex]dU=TdS +\mathcal{F}dL[/itex]

[itex]\left(\frac{\partial U}{\partial L}\right)_T=T\left(\frac{\partial S}{\partial L}\right)_T +\mathcal{F}\left(\frac{\partial L}{\partial L}\right)_T[/itex]

[itex]\left(\frac{\partial U}{\partial L}\right)_T=T\left(\frac{\partial S}{\partial L}\right)_T +\mathcal{F}[/itex]

Then using the maxwell relation [itex]\left(\frac{\partial \mathcal{F}}{\partial T}\right)_L=\left(\frac{\partial S}{\partial L}\right)_T[/itex]

We have

[itex]\left(\frac{\partial U}{\partial L}\right)_T=T\left(\frac{\partial \mathcal{F}}{\partial T}\right)_L +\mathcal{F}[/itex]

Using the equation of state we then have [itex]\left(\frac{\partial U}{\partial L}\right)_T=2\mathcal{F}[/itex]
 
  • #5
OK. This is good. But, check your signs in the Maxwell relation. Note ##\mathcal{F}## for the rubber band corresponds to ##-P## for a usual thermodynamic system.
 
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  • #6
So its zero which makes the next part trivial, thanks for the help.
 

Related to How Do You Derive the Energy Equation for a Rubber Band?

1. What is the relationship between heat and rubber band elasticity?

The thermodynamics of rubber bands is based on the principle of entropy, which states that heat increases the disorder or randomness of a system. In the case of rubber bands, when heat is applied, the molecules within the rubber band vibrate more rapidly, increasing the disorder and causing the rubber band to become more elastic.

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

The length of a rubber band affects its thermodynamic properties because it determines the amount of internal energy and entropy within the rubber band. A longer rubber band has more internal energy and higher entropy, making it more elastic and easier to stretch.

3. Can a rubber band be stretched indefinitely without breaking?

No, a rubber band has a maximum limit to its elasticity. As it is stretched, the molecules within the rubber band are pulled apart, and at a certain point, they can no longer return to their original position. This results in the rubber band reaching its maximum elasticity and eventually breaking.

4. How does temperature affect the thermodynamics of a rubber band?

Temperature has a direct impact on the thermodynamics of a rubber band. As the temperature increases, so does the kinetic energy of the molecules within the rubber band, causing them to vibrate more rapidly and increasing the rubber band's elasticity. On the other hand, a decrease in temperature results in a decrease in elasticity due to the decrease in molecular motion.

5. Is the thermodynamics of a rubber band affected by its color or composition?

The color and composition of a rubber band do not significantly impact its thermodynamic properties. However, the composition, specifically the type and amount of polymer used, can affect the strength and elasticity of the rubber band. Generally, darker-colored rubber bands tend to be stronger and more elastic than lighter-colored ones due to the higher concentration of carbon black used in their production.

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