chem456_ch3lecture - U and H for non-ideal gasses Need to...

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1 U and H for non-ideal gasses • Need to follow the two energy terms (U and H) as functions of P,V and T. • Need to use calculus to find the appropriate derivatives and understand the terms that define the energy changes • Use the “Thermodynamic Equation of State” • Apply the EoS to relate U and H and relate Cp and Cv • Understand the Joule experiment • Understand the Joule-Thompson experiment
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2 Chapter 3: Following Changes in State Functions Using Mathematics to help us keep track of changes in state functions. In one dimension, a relation in calculus: () ( ) 2 1 21 Chain Rule Indefinite Integral Definite Integral x x df ff x d f d x dx df fd x dx df x f x d x dx ⎛⎞ == ⎜⎟ ⎝⎠ = Δ= = This is the main method we have to solve problems in thermodynamics.
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3 Apply to Newton’s Mechanics • Distance is a function of time • Velocity is the derivative of position in time () () () 2 1 t t df f xt v vt x d t dt == = Δ = Suppose the velocity is a constant (object in free space) oo vv xv t = Δ Suppose the velocity increases linearly with time (object falling under gravity) 2 1 22 21 11 t t v t gt x gtdt x g t t g t t t = Δ = = + Δ
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4 Notation to include multiple variables • Change the derivative symbol from • Use subscripts to tell which variables are held constant (when taking derivatives) • Can write changes so that only one variable changes at a time. d →∂
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5 () .. , eg RT Pf V T n V == VT PP dP dT dV TV ∂∂ ⎛⎞ =+ ⎜⎟ ⎝⎠ The Mathematical Properties of State Functions The above notation already implies that n is fixed. If n can vary also, we can generalized the equations further. ,,, Vn Tn PPP dP dT dV dn n ∂∂∂ + These statements are always correct.
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6 Evaluating Partial Derivatives () 1 VT V V PP dP dT dV TV nRT dn R T Pn R P V TT V d T V T ∂∂ ⎛⎞ =+ ⎜⎟ ⎝⎠ = == = Demonstrate derivatives, called partial derivatives, with the I.G. E.o.S. 1 2 T T nRT P Pd V n R T P V nRT VV d V V V = = Use Reciprocal rule 1 T T P V = =−
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7 Derivates in different order 2 2 2 2 2 T T V V V T RT V V RT P PR V TV T V T T V RT V T R PP R V VT V T V V V ⎛⎞ ⎜⎟ ⎝⎠ ∂− ⎜⎟ ⎜⎟ ∂∂ ⎝⎠ ⎝⎠ = === == Demonstrates that the order does not matter for state functions () ( ) 22 ,, or V T fV T T V T T V V T
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8 State functions can be integrated over the various variables. For example the pressure is a function of the temperature and volume. However: When we have a process where only the temperature varies: For Ideal Gas f i T final initial V T P P PP d T T nR PT V ⎛⎞ Δ= = ⎜⎟ ⎝⎠ Δ Integrate State Function This equation says that P can be expressed as an infinitesimal quantity, dP , that when integrated depends only on the initial and final states. df is called an exact differential . U is also an example of a state function.
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chem456_ch3lecture - U and H for non-ideal gasses Need to...

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