Adiabatic Stretching of a Rubber Band and the First Law of Thermodynamics

[ryanwilk]

I need to show that adiabatic stretching of a rubber band causes an increase in temperature.

I've managed to reduce the 1st Law of Thermodynamics to dU=kLdL.

k,L and dL are all positive so dU is positive - the total internal energy increases.

But does this immediately imply that the temperature also increases?

Any help would be appreciated.
Thanks.

 

[ryanwilk]

(I've basically done:
dU = dQ + dW
dQ = TdS and dW=-PdV+fdL
=> dU = TdS-PdV+fdl
Assume constant volume => dV=0
f=kL
=> dU = TdS+fdl
S is constant in an adiabatic process so dS=0
=> dU = kLdL)

 

[Mapes]

I'd start with (\partial T/\partial L)_S and start applying differential identities, Maxwell relations, etc., to get it in terms of derivatives whose sign you know.

For example, are (\partial S/\partial T)_L, (\partial L/\partial T)_F, (\partial L/\partial F)_T positive or negative for an elastomer?