UNIVERSITY OF OSLO Obligatory assignment 2: FYS2160, Thermodynamics and statistical physics, Fall 2018 Due: October 08. 2018

1 Elastic rubber band

For an elastic rubber band of length L(T,P ) at temperature T and under a tension P , it is found experimentally that the tension in the band changes with T and L as " # ∂P  aL L 3 g(L) = = 1 − 0 ∂T L L0 L " # ∂P  aT L 3 f(T,L) = = 1 + 2 0 , (1) ∂L T L0 L where L0 is the length of the un-stretched band (independent of temperature) and a > 0 is an empirical constant. 1.1 Verify that P (T,L) is a well-defined function, that is

 ∂ ∂P    ∂ ∂P   = (2) ∂T ∂L T L ∂L ∂T L T by checking the Maxwell relation  ∂f   ∂g  = (3) ∂T L ∂L T

1.2 Show that f(T,L) = T g0(L) (g0(L) = dg/dL).

1.3 Derive an equation for state P (T,L), that is a relation between tension P , temperature T and length L for the elastic rubber band system.

The rubber band obeys the thermodynamic relations that are analogous to those of a gas. The work done on the elastic band when its length increases by an infinitesimal amount dL is δW = +P dL, hence the first law has the thermodynamic identity

dU = T dS + P dL. (4)

1.4 Explain why Helmholtz free energy F (T,L) is the appropriate thermodynamic potential for this problem. Find the thermodynamic identify for F (T,L). 1.5 Show that the above thermodynamic identity leads to the following Maxwell relation ∂S  ∂P  = − (5) ∂L T ∂T L

1.6 From the equation of state P = T g(L), it follows that (∂P/∂T )L = P/T . Using this together

1 with the above Maxwell relation, show that the internal energy of the band depends only on T , just like for an ideal gas. (show that (∂U/∂L)T = 0)

1.7 The heat capacity at constant length L is then related to changes in the internal energy at fixed L as dU = CLdT . Assume that CL is constant. Suppose the band is stretched adiabatically and reversibly (Qrev = 0), i.e. isentropically (dS = 0), from L0 at an initial temperature Ti to the final length Lf . What is its final temperature Tf ?

1.8 The stretched rubber band with the final length Lf is released, so that it contracts freely to its equilibrium unstretched length L0. If no heat and no work is exchanged with its surrounding during this contraction, find the changes in its temperature and entropy. What is the analogue process for a gas?

2 Maxwell relations and response functions

The difference in the heat capacities CV and CP of any substance can be expressed in terms of temperature T , volume V and the response functions of the substance. The other response functions are: Isothermal compressibility κT measures the relative volume change by changing pressure at fixed temperature 1 ∂V  κT = − (6) V ∂P T Thermal expansion coefficient α measures the relative volume change by changing temperature at fixed pressure 1 ∂V  β = (7) V ∂T P 2.1 Regarding the entropy S as a function of T and V , we have that  ∂S   ∂S  dS = dT + dV (8) ∂T V ∂V T and volume V as function of T and P ∂V  ∂V  dV = dT + dP (9) ∂T P ∂P T show that the heat capacities CP and CV are related as ∂P  ∂V  CP = CV + T (10) ∂T V ∂T P

2.2 Apply this formula to the ideal gas.

2.3 Using the identity ∂V  ∂P  ∂V  = − (11) ∂P T ∂T V ∂T P

2 show that T V β2 CP − CV = (12) κT

2.4 Find the expressions for β and κT for the ideal gas.

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