Physical Chemistry
Lecture 19
Elastic Networks
Elastic network
Do work by pulling on the network
dwstretch
=
fdl
Also receives energy by heat transfer and
volume change
dU = dq − Pext dV + f ext dl
Reversible process
dU
= TdS
− PdV
+
fdl
Thermodynamics
Assume constant volume upon stretching
dU
= TdS
+
fdl
Free-energy function
dA = − SdT
+
fdl
Maxwell relation
 ∂f 


 ∂T l
 ∂S 
= − 
 ∂l T
Energy change with temperature
Substitution for dS gives the energy as a
function of T and l
dU
 ∂S 
= T
 dT
 ∂T l

+ f

 ∂S  
+ T   dl
 ∂l T 
 ∂S 
= T
 dT
 ∂T l

+ f

 ∂f  
− T
 dl
 ∂T l 
Ideal elastic network
 ∂U 


 ∂l T
= 0
⇒
 ∂f 


 ∂T l
=
f
T
Adiabatic change of length
How does temperature change when the
length changes, while the entropy is held
constant?
 ∂T 


 ∂l  S
 ∂T   ∂S 
= − 
 
 ∂S l  ∂l T
= −
T  ∂S 
 
Cl  ∂l T
Since the derivative of S with l is negative,
the derivative of T with l must be positive.
A quick drop in length is accompanied by a
temperature drop!
Temperature dependence of
length
What happens to an extended network that is
heated?
 ∂l 


 ∂T  f
= ?
Derivative positive ⇒ length increases as
temperature goes up
Derivative negative ⇒ length decreases as
temperature goes up
Evaluation of derivative
By the cyclic rule
 ∂l 


 ∂T  f
 ∂l
= − 
 ∂f
  ∂f 
 

T  ∂T l
By the reciprocal rule and the
assumption of an ideal network
 ∂l 


 ∂T  f
 ∂f 


∂T l

= −
 ∂f 
 
 ∂l T
f /T
= −
 ∂f 
 
 ∂l T
Hooke’s law
Treat the network as a simple spring
f (l , T ) = k (l − lo )
k is the Hooke’s-law constant
 ∂f 
 
 ∂l T
= k
Evaluation of derivative
Temperature derivative for a Hookean
network
 ∂l 


 ∂T  f
f /T
= −
 ∂f 
 
 ∂l T
k (l − l0 ) / T
= −
k
(l − l0 )
= −
T
Evaluating length change
Integration allows a prediction of the length
Separate variables
1  ∂l 


l − l0  ∂T  f
1
= −
T
Integrate over temperature
1  ∂l 
∫1 l − l0  ∂T  f dT
2
2
1
∫1 l − l0 dl
2
1
dT
T
1
= −∫
2
1
dT
T
1
= −∫
 l2 − l0 
 T2 


= − ln 
ln

 T1 
 l1 − l0 
Temperature dependence of
length of a network
Rearrange equation
l2
− l0
T1
(l1 − l0 )
=
T2
The higher the temperature, the shorter
the network
Summary
Thermodynamics of a network shows
the predictive ability of theory
Adiabatic processes detected by change
in temperature
Hookean model predicts that a network
subject to a constant force will contract
when heated