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Unformatted text preview: Summary of last lecture • Dependence of density of states on external parameters (formulation of the problem and announcement of the result) • Generalized forces • Laws of Thermodynamics • Exact (TD potentials) and inexact (work and heat) differentials, functions and nonfunctions of state. • Entropy and heat Key Concepts and Lessons: a) Equation of state for an ideal gas b) Heat capacity at constant V and constant p c) Thermodynamic potentials and transfromations d) Stability conditions Reading: Reif, Ch5 Equation of State • Various thermodynamic potentials (such, E,S,T,V,p etc) may not all be independent but connected through a relation which is called equation of state . Consider, e.g. a very important example of ideal gas. We already established a relation E = 3 N 2 kT What about other quantities such as p,V,T? Any connection? (You might recall your younger years when you heard about that from kinetics theory derivation) Recall our beauty: X α = 1 β ∂ ln Ω ∂ x α where X α = − ∂ E r ∂ x α is a generalized force conjugated to the parameter x α Now use V as a parameter and p as a generalized force. Recall expression for entropy of ideal gas that we derived earlier: Ω E ( ) ~ 1 N ! 2 π m h 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3 N /2 V N E 3 N /2 − 1 ( ) S ( E ) = CONST + N ln V + 3 N 2 ln E Equation of state for an ideal gas • Now we apply our relation to get p = 1 β ∂ ln Ω ∂ V = N β V = NkT V A very familiar relation which we derived entirely within the statistical mechanical paradigm! Another interesting example of equation of state Is for vanderWaals gas which we will derive later. Unlike in an ideal gas, in the vdW gas particles do interact and it substantially affects its properties p + a V 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ V − b ( ) = NkT Heat capacity • Is a measure of how much heat needs to be transferred to a system in order to increase its temperature by 1 degree, or more generally by dT Where y denotes specific conditions at which heat is transferred because we know that Q is not an exact differential and we have to be careful about conditions at which heat transfer takes place. Of particular interest are heat capacity at constant volume and heat capacity at constant pressure . At constant volume work W=0 (no moving parts, i..e no pdV) and C V = dQ dT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ V = dE dT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ V C y = dQ dT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ y = T ∂ S ∂ T ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ y Heat capacity of monoatomic ideal gasconstant volume • A) C V . In this case no work is done and all heat goes to change of internal energy. Heat capacity is easy to evaluate: C V = dQ dT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ P =...
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This note was uploaded on 05/04/2010 for the course CHEM 161 taught by Professor Shaklovich during the Spring '10 term at Harvard.
 Spring '10
 Shaklovich

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