12. MEASUREMENT OF THE POISSON CONSTANT THE METHOD CLEMENT-DESORMES METHOD
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1 1. MEASUREMENT OF THE POISSON CONSTANT THE METHOD CLEMENT-DESORMES METHOD ASSIGNMENT 1. Measure the rato of C P CV. Calculate the error of the measurement THEORETICAL PART Substances dffer from one to another n the quantt of heat needed to produce a gven rse of temperature n a gven amount of the gas. The amount of heat Q requred to change the temperature of a sstem s found to be proportonal to the mass m of the sstem and the temperature change T. Ths can be epressed b the equaton Q = mc T (10.1) where c s a quantt characterstc of the materal and t s called the specfc heat. The values of the specfc heats depend to some etend on the temperature, but for a small temperature changes c can be often consdered constant. The specfc heat depends also on the knd of processes heatng. In eq.10.1 we assumed the process was carred out at the constant pressure-atmospherc pressure. For such a process we usuall call c the specfc heat at constant pressure. and t s denote b the smbol c P. If the volume of the materal s kept constant, then specfc heat s called as specfc heat at constant volume and t s denote b the smbol c V. For the solds and lquds the dfference between c P and c V s small and we can wrte 1 Q 1 du cp = cv = = (10.) m T m dt However the dfference of gases between c P and c V are qute dfferent Ths ma be eplan n terms of the frst law of thermodnamcs and the knetc theor of gases We ntroduce the molar heat capact at constant volume as dq Cv = 1 (10.) n dt V and the molar heat capact at constant pressure as dq CP = 1 (10.3) n dt P whch are defned, as the heat requred rasng 1mol of the gas b 1K at constant volume and constant pressure. We shall magne that the gas s gong from one state to another state quasstatstcall, b whch we mean that the process s carred out etremel slowl through a successon of nfntesmall close equlbrum states. The change n nternal energ for an deal gas can be epressed as U = ncv T (10.4) where C v s the molar heat capact of the gas at constant volume, n s the number of moles and
2 T s the change n the temperature between two states of a gas. In the lmt of dfferental changes we can use the frst law of thermodnamcs to epress the molar heat capact n the form du Cv = 1 (10.5) n dt The nternal energ of the monatomc gas equals U = 3 nrt (10.6) Insertng ths equaton nto eq.10.5 gves the value of the molar heat capact at constant volume as C = 3 v R (10.7) where R s the unversal gas constant Now suppose that the gas s transferred to the sstem at constant pressure. Let the temperature ncreases b T. The heat must be transferred to the gas. Its value s gven b Q = ncpt. Snce the volume ncreases n ths process, the work done b the gas s W = PdV. Applng the frst law of thermodnamcs we get U = Q W or ncv T = ncp T nr T (10.8) If ou can see from ths epresson the part of the heat transferrng to the sstem ncreases the nternal energ of the sstem and second part s used to do eternal work b movng a pston. From ths epresson follows the ver mportant relaton between the molar heat capact at constant pressure and constant volume as C p = Cv + R (10.9) If ou can see from ths epresson the molar heat capact at constant pressure s grater than the molar heat capact at constant volume b an amount of the unversal gas constant. We ntroduce the a new phscal constant CP γ = CV Ths constant s dmensonless quantt and t s called the Posson constant. Its value depends on the number of atoms n the molecule. For an monatomc gas, for eample, the value of ths constant s 5 γ = (10.10) 3 The values of C p and χ are n ecellent agreement wth epermental values for monatomc gases. The nternal energ and hence the molar heat of a comple gas must nclude contrbutons from the rotatonal and vbraton moton of molecule. The rotatonal and vbraton moton of molecules wth structure can be actvated b collsons and therefore are coupled to the translaton moton of molecules. The statstcal mechancs has shown that for a large number of partcles the avalable energ s, on the average, shared equall b each ndependent degree of freedom. The equpartton theorem states that at equlbrum each 1 degree of freedom contrbutes, on the average, kt of energ per molecule. Then for the degree of freedom we have
3 Cv = R (10.11) and Cp χ = = + (10.1) Cv For a datomc molecule s the stuaton shown n Fg.10.1 v translaton moton rotatonal moton vbraton moton of the centre of mass about varous as along the molecular as For ths molecule we can neglect the rotaton about - as snce the moment of nerta and the rotatonal energ 1 Iω about ths as are neglgble compared wth those assocated wth the and -as. Thus there are fve degrees of freedom- three degrees of freedom assocated wth the translaton moton and two assocated wth the rotatonal moton. Then from eq. (10.11) we have C = 3 v R (10.13) Therefore, the value of Posson s equaton for datomc molecules equals 5 χ = = 140. (10.14) If an deal gas undergoes an adabatc epanson or compresson, the frst law of thermodnamcs together wth the equaton of state shows that χ PV = const (10.15) Equaton s called the Posson equaton for a adabatc process. Usng ths equaton for a gas that transfers from the ntal state to another fnal state we have χ χ P V = bvf (10.16) From ths equaton follows δ Vf P = V (10.17) b The rato of the ntal and fnal volume ma be calculated b the usng the Bole law as V f P = (10.18) V P f Then the rato of heat capactes χ can be wrote as log P logb χ = (10.19) log P log Pf
4 THE METHOD - PRACTICAL PART Clement and Desormes n 1819 proposed the smple method of measurement two mportant constants of gases: the molar heat capact C p at the constant pressure and the molar heat capact C v of the gas at the constant volume. The devce for ths measurement s shown n Fg.10.. In a bulb B s closed a gas. The open-tube manometer measures the pressure nsde the bulb. Then the pressure of gas (ar) n bulb s equal h B K P = b + ρgh (10.0) where b s atmospherc pressure, ρ s the denst of the lqud, h s the dfference between the heghts of the lqud.n the arms of U-manometer and g s acceleraton of gravt The method s based on the measurng the pressure P 1 of the ar n the bulb untl enhanced the pressure through the stopcock K 1 and the pressure of the ar f the pressure n the ar n the bulb decreases after the stopcock K s quckl turn round. Pump the ar through the stopcock K 1 nto the bulb and read the dfference between the columns of the lqud n manometer. Quckl turn round the stopcock K.If the ar n the bulb s n thermal equlbrum measure the dfference between the column of lqud n manometer agan. Note the process must be adabatc,.e. no heat enters or leaves the sstem ( Q = 0 ). An adabatc process can be acheved b the performng the process rapdl. MEASUREMENT Apparatus: Clement-Desormes apparatus, barometer, thermometer. We shall measure the Posson constant of the ar. The ar conssts of 78% of ntrogen, 1% of ogen and 1% of rare gases. All gases are datomc ones. Therefore, the Posson constant wth respect to eq has the value of Measure the atmospherc pressure b b the barometer. Measure the temperature t. Measure the pressure of the ar n the bulb b the method that s descrbed above. Repeat these measurements a few tmes and record them nto Tab CALCULATION Usng eq.10.0 calculate the pressure P and P f for ever measurement. Usng eq calculate the Posson constant γ for each par of P and P f. Remember that the pressure P s the pressure of the ar after the pumpng the ar nto the bulb and P f s the pressure of the ar f the ar s loose from the bulb. Calculate the percentage error of measurement as
5 γ.100% γ t where γ = γ m γ t, γ t = Analse the source of error n ths eperment
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