The Joule Expansion Experiment

For a closed system, the state function U is determined by T and V alone: ∂UU ∂ dU=+ dT  dV ∂∂TVVT  ∂U =+CdTV  dV ∂V T Joule tried to measure this partial derivative.

1. Gas in A, T vacuum in B. AB2. Open valve. 3. Any change in T?

He found no change in temperature when the gas expanded from VA to VA+VB, i.e. q = 0. Also, no work was done (free expansion), so w = 0. Conclusion: ∆U = 0 and hence ∂U = 0 orUUT= () only ∂V T Strictly only true for ideal gases.

Not true for liquids and solids, but since ∆UUVV≈∂/ ∂ ∆ ( )T and ∆V is very small, the effect of ∆V on U is usually ignored.

Paul Percival CHEM 360 Spring 2004 1/25/2004 Pressure Dependence of Enthalpy

For a closed system, the state function H is determined by T and P alone: ∂HH ∂ dH=+ dT  dP ∂∂TPPT  ∂H =+CdTP  dP ∂P T For ideal gases HUPVUnRT=+ =+ ∂∂HU = = 0 ∂∂PPTT

Not true for real gases, liquids and solids! dH=+ dU PdV + VdP ∂∂HU  CPV dT+=+++ dP C dT  dV PdV VdP ∂∂PVTT  At fixed temperature, dT = 0 ∂∂∂∂HUVV  =++PV  ≈ V ∂∂∂∂PVPPTTTT  small for liquids and solids Investigate real gases at constant enthalpy, i.e. dH = 0 ∂∂HT  ⇒ =−CP  ∂∂PPTH 

Paul Percival CHEM 360 Spring 2004 1/25/2004 The Joule-Thomson Experiment ∂T Joule-Thomson Coefficient µ=JT  ∂P H

Pump gas through throttle (hole or porous plug) P1 > P2

P1 Keep pressures constant by moving pistons. Work done by system

−=wPVPV22 − 11 Since q = 0 P2

∆=UU21 − U == wPVPV 1122 −

⇒ UPVUPV222111+=+

HH21= constant enthalpy Measure change in T of gas as it moves from side 1 to side 2. ∆T µ=JT  ∆P ∆P→0 A modern, more direct experiment ∂Hq  uses similar apparatus but with a = lim ∆→P 0  heater to offset the temperature drop. ∂PPT ∆ 

Paul Percival CHEM 360 Spring 2004 1/29/2007 The Linde Refrigerator

For most gases at room temperature µJT > 0 so sudden (adiabatic) expansion results in a drop in T. This is the basis of operation of the Linde refrigerator and gas liquefaction.

cooling fins cold gas

expansion compressor nozzle

In general, µJT depends on T and can even change sign. inversion temperature T ∂T > 0 lines of ∂P H constant ∂T enthalpy < 0 ∂P H P

Paul Percival CHEM 360 Spring 2004 1/30/2005 The Molecular Interpretation of U

What is U? ∆U can be related to thermochemical observables: T ∂U 2 CUCdT=∆=, for ideal gases VV ∫T ∂T V 1 T UT()−= U ()0 CdT ∫0 V Can U be calculated from molecular properties? UuNu==<>∑ ignoring intermolecular forces molecules Classically, is given by the Equipartition Law: “The average energy of each different mode of motion of a molecule is ½kT.” k = Boltzmann constant = R/L =1.381 × 10-23 J K-1 Implicit in this “law” is the concept that there is no coupling between molecular modes of motion.

Total: u =εtr +ε rot +ε vib ()+εel 11112222 Translation: ε=tr 2222mv = mx + my + mz 111222 Rotation: ε=rot 222IIIx ω+ω+ωxyyzz 1122 Vibration: ε=+=vib TV22 mxkx + per mode 36n − for non-linear molecules no. of vib. modes =  35n − for linear molecules

Paul Percival CHEM 360 Spring 2004 1/25/2004 The Molecular Interpretation of U (cont.)

When quantum effects can be ignored, the average energy of every quadratic term in the energy expression has the same value, ½kT.

3 ∂U Monoatomic gases <>=ukTC2 ,1.5V = = R ∂T V 3 Diatomic gases < ukTCR>=( 2 +11 +) ,V = 3.5

3 Linear polyatomics < u >=( 2 +13 + nkT− 5)

e.g. for nC= 3,V = 6.5R

33 Non-linear molecules < u >=( 22 + +3n −6)kT

e.g. for n ==3,CV 6.0R Experimental observations do not agree with these predictions, except for monoatomic gases. In particular: ! The predicted values are too high. ! Experimental values are temperature-dependent.

Accurate calculations are provided by statistical mechanics, where it can be shown that the equipartition principle only holds in the limit of high temperature, specifically for the condition kT >> ∆E, where ∆E is the appropriate spacing of energy levels. Equipartition ≡ free transfer of energy between modes

Paul Percival CHEM 360 Spring 2004 1/25/2004 Temperature Dependence of CV ∂U CUUNV =−=<ε>+<ε>+, where 0trrot()… ∂T V tr rot vib el =+CCVV + C V + C V CRtr =1.5 for all T V

rot R – linear CkT=∆ε for rot V {1.5R – non-linear C rot = 0 at very low T, where < ε>→0 V rot CRvib = for each vibrational mode, at high T V C el = 0 almost always, since electronic excitation V takes great energy e.g. for a diatomic molecule: → two atoms CV R 7 t + r + v 2 2t 5 t + r 2 3 t 2

0 0 T

Paul Percival CHEM 360 Spring 2004 1/25/2004 Temperature Dependence of H

Interactions between molecules also contribute to the heat capacity of real systems. In particular, first-order phase changes often involve large energy changes. These are usually measured at constant pressure and expressed as enthalpies.

gas

H ∆H v liquid solid 2 ∆Hf

solid 1 ∆Hs-s 0 0 T

In general, HabTcT=+ +2 +… for each phase ∂H CbcTP ==++ 2 … ∂T P

Paul Percival CHEM 360 Spring 2004 1/31/2004