Problem Set 5 Ch21c Spring 2014 TA Michiel Niesen Ruijie Kelly Zhang mniesen caltech edu rzzhang caltech edu Due May 22 2014 5pm in Michiel s mailbox in Noyes Reminder Write how long the problem set took on the front page of your problem set Problem 1 Quantum Harmonic Oscillator and the Third Law A quantum harmonic oscillator has energy E n 1 2 n 0 1 2 a Calculate the canonical partition function b Calculate the Helmholtz free energy A c Calculate the entropy S d Show that this system follows the third law of thermodynamics Problem 2 Isothermal Decrease in Pressure a Calculate the differential form for the Gibbs free energy G U T S P V What are the natural variables of G b Find a Maxwell relation relating the entropy to the volume c For an ideal gas calculate the change in entropy S for a reversible isothermal process where the pressure decreases from P to P P Assume P is positive d Relate the initial pressure P with a volume V and the final pressure P P with a final volume V V Show that the derived entropy in part c is the same as the change in entropy found in class for an isothermal expansion from V to V V What does this suggest about the information contained in different thermodynamic ensembles 1 e Now for the isothermal expansion of V to V V calculate the change in Helmholtz free energy A f Using your result from part e calculate the relative probability of the ideal gas occupying the volume V V compared to occupying the volume V Problem 3 Stability From Chandler Introduction to Modern Statistical Mechanics 2 9 An experimentalist claims to find that a particular gaseous material obeys the conditions i ii iii iv p V p T V T V 0 T 0 V 0 T 0 S a Identify which of these inequalities is guaranteed by stability b Identify which pair of inequalities is inconsistent with each other and demonstrate why they are inconsistent Problem 4 Thermodynamics of a Rubber Band An experimentalist observes a rubber band that is being pulled by a tension f heats up when it is stretched adiabatically That is T 0 f S Given this fact it can be determined whether a rubber band will contract or expand when it is cooled at constant tension T a Starting from the given inequality show that 0 L S 2 b Use the triple product rule to show that S L 0 T c Use a Maxwell relation and the triple product result to determine the sign of L T f Does this imply the rubber band will contract or expand when it is cooled d Show that Cf CL T f L T L T 2 0 f Based on this will a rubber band held at constant tension or at constant length have a larger increase in temperature when the same amount of heat flows into the rubber bands For the remainder of the problem consider a particular type of rubber band has the equation of state l f T where l is the length per unit mass f is the tension T is the temperature and is a constant e For this equation of state calculate Cf Cl f What is the differential form of the reversible work for the extension of a rubber band Using this write the differential form for a change in the internal energy U g Construct Legendre transforms of the the internal energy that are natural functions of T l and T f h Find Maxwell relations relating the entropy to the tension and relating the entropy to the length per unit mass Cl i Compute where Cl is the constant length heat capacity l T j Calculate the change in entropy S for isothermally increasing the tension from f to f f Problem 5 Equations of State From Kardar Statistical Physics of Particles 1 4 3 The equation of state relating pressure to volume and temperature constrains the form of internal energy as in the following examples Assume N is fixed for simplicity a Starting from dU T dS P dV show that the equation of state P V N kB T in fact implies that U can only depend on T b What is the most general equation of state consistent with internal energy that depends only on temperature c Show that for a van der Waals gas that behaves according to the following equation of state CV is a function of temperature alone N kB T P a V Nb 4 N V 2