Download VEE-HC-12 - Háskóli Íslands

Háskóli Íslands
Raunvísindadeild,
Verklegar æfingar í eðlisefnafræði / EFN509G
Varmarýmdarhlutföll lofttegunda /
Heat capacity ratios for gases
INNGANGUR
Varmarýmd (C / kJ mol-1 K-1) er sú varmaorka sem þarf til að hita upp eitt mól efnis (q ( J
mol-1 eða kJ mol-1) um eina gráðu:
C = dq/dT
Varmaorka per mól efnis við fast rúmmál (qV) er jafngilt innri orku breytingu efnisins (dU).
Tilsvarandi varmarýmd er táknuð sem CV.
Varmaorka per mól efnis við fastan þrýsting (qP) er jafngild enthalpy (vermin-) breytingu
efnisins (dH). Tilsvarandi varmarýmd er táknuð sem CP.
CV = dqV/dT
CP = dqP/dT
Varmarýmdarhlutfall ()
 = CP /CV
-fyrir kjörgas er háð hljóðhraða (c), mólmassa (Mw) og hita (T) skv.
 = (c2Mw)/(RT)
(1)
Varmarýmd gass við fast rúmmál (CV) má þátta í varmarýmdargildi vegna færsluorku (e.
Translational energy, CV(T)), snúningsorku (CV(J)) og titringsorku (CV(v)):
CV = CV(T) + CV(J) + CV(v)
Fyrir kjörgas, fast efnatengi (e. rigid rotor) og kjörsveifil (e. harmonic oscillator) gildir:
CV(T) = (3/2)R
(2a)
CV(J) = R; línuleg sameind / CV(J) = (3/2)R; ólínuleg sameind
(2b)
CV(v) =
h
Rx 2 e x
per sveifluform(e vibrational mod e) i; x  i
x
2
k BT
(e 1)
(2c)
FRAMKVÆMD
Ákvarða skal varmarýmdarhlutfall lofttegunda skv. Hljóðhraðamælingum (sbr. Jafna (1) hér
ofar) og niðurstöður bornar saman við fræðilega útreiknuð gildi (sbr. jöfnur (2 a-c) ofar).
-er tvíþætt (A og B), háð mæliaðferðum:
A) Hljóðhraðamæling fyrir fasta hljóðbylgjutíðni fyrir eitt efni (köfnunarefni (N2(g)).
B) Hljóðhraðamælingar (og Fouriergreining) fyrir hljóðbylgjutíðnisvið og fourier greining
fyrir þrjú efni (N2(g), CO2(g), Ar(g))
A) Hljóðhraðamæling fyrir fasta hljóðbylgjutíðni fyrir eitt efni (köfnunarefni (N2(g)):
-sbr. lýsing í GNS7 / GNS8* æfing 3B
-Sjá einnig:
Introduction:
The velocity of sound in a gas is estimated by measuring the wavelength of standing waves with a well defined
frequency in a tube filled with the gas. This is procedure B in experiment 3 in GNS7/GNS8. As discussed in the
introduction to the method in GNS7/GNS8, the speed of sound is proportional to the square root of the heat
capacity ratio Cp/Cv. The experiment can therefore be used to give an estimate of the heat capacity of gas
molecules. The heat capacity at low temperature (low being room temperature here) is one of the the four main
experimentally measured quantities that showed discrepancy with the predictions of classical physics at the
beginning of the 20th century and led to the development of quantum mechanics.
Procedure:
The heat capacity ratio of the gas N2 will be measured.
Read carefully the instructions for working with gas cylinders in GNS7/GNS8 (in the chapter on "Miscellaneous
Procedures") before coming to the lab.
Connect the wave generator to the frequency monitor and the scope. The "strong" output from the wave
generator should be connected with the freqeuncy monitor, and the "weak" output should be connected with the
scope. Set the frequency generator to 1000 Hz and record carefully the actual frequency. Set the filter on and
average over 1 second on the frequency meter. Display a Lissajous figure on the scope to determine the distances
between speaker and microphone that correspond to in phase and out of phase coherence.
Start with the microphone pulled out far from the speaker. Run N 2 gas through the tube for at least 10 minutes
before starting the measurements. You can monitor the gas flow by dipping the plastic hose into the silicon oil.
Pull the hose up before measuring because the formation of bubbles in the oil causes noise in the measurement.
Use the metal marker near the microphone to read the position accurately. Measure the positions corresponding
to in-phase and out-of-phase interference as the microphone is brought towards the speaker. Repeat the
measurements as the microphone is pulled out again. Be careful not to pull the microphone out so fast that air
gets sucked into the tube. This can be monitored by the gas bubbles in the silicon oil. Repeat the measurements
as the microphone is pulled in and out again to check reproducability. If the same results are not obtained, repeat
the measurements. After reliable measurements have been obtained for this frequency, set the frequency
generator to 1500 Hz and measure again at least two rounds of pulling the microphone in and then out again.
Which frequency gives more consistent measurements? Record the temperature of the gas in the tube and obtain
the atmospheric pressure from the web page of the meteorology institute (Vedurstofa Islands).
Analysis:
Make a table of the positions corresponding to in-phase and out-of-phase interference. List how many half
wavelengths each of these positions correspond to. Calculate the best estimate of the wavelength by taking an
average. Calculate the speed of sound in the gas.
B) Hljóðhraðamælingar (og Fouriergreining) fyrir hljóðbylgjutíðnisvið og fourier greining
fyrir þrjú efni (N2(g), CO2(g), Ar(g)).
Sbr. lýsingu í C. Steel, T. Joy and T. Clune, J. Chemical Education, 67(10), 883 – 887, (1990).
Furthermore: Use a fixed long distance between the microphone and the detector in this part
of the experiment, ca 1 meter; NB!: the effective distance (L) between the microphone and the
detector should be determined from experimental data for A (N2(g))
Tölvusöfnun gagna (sjá nánar http://www3.hi.is/~agust/kennsla/ee09/vee09/VEE-HCc09.pdf ):
Í upphafi:.Fáðu kennara til að setja upp Labjack og magnara tengingu milli tölvunnar og
hljóðnemans. ATH: Labjack magnarinn þarf að vera stilltur á 1:1000 mögnun. Mikilvægt er
að tengja LabJack USB tengi frá mæli í efra USB portið framan á tölvunni. Útbúið möppu til
að vista gögn í: Smellið á „My Documents“, Opnið möppuna „nemar VEE“ og útbúið ykkar
eigin möppu (t.d. „Jón og Gunna“). Nota skal „FFT-gases“ forritið sem er í samnefndri
möppu á skjáborði.
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Smella á „FFT-gases“
Mælitíðnin (scan rate) ætti að vera meira en 10 kHz (þarf að vera minna en 50 kHz).
Fjöldi mælinga (FFT samples) má vera einhver “skynsamleg tala” (>10). Eftir hverja
mælingu birtast spennugildi ásamt Fourier ummyndun.
 Ýtið á „START“ til að hefja mælingu:
Að lokinni gagnasöfnun fyrir n (hér 25) mælingar birtist:
a)
Síðasta mæling á spennugildi frá hljóðnemanum efst til vinstri
b) Fourier greining (FFT) spennugildanna (a) efst til hægri.
c)
Meðaltal (FFT) n mælinga neðst.
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Vistið mæligögn í mæliskrá. Gefið mæliskránni nafn (t.d. FFTmaeling1)
Unnt er að stækka myndir til að sjá form mælingar betur, t.d. breyta hæstu tölu á x –ás
í 3000:
Endurtakið mælinguna fyrir mismunandi upphafsgildi (scan rates / FFT samples) til að
besta mælingu.
ATH: Aftengið USB tengi LabJacksins að mælingum loknum.
Flytjið nú gögn yfir í IGOR Pro til frekari úrvinnslu. Stundum getur verið mismunur
á notkun punkta og komma í gagnaskrám forrita. Þá getur verið hentugt að breyta
skjölum með t.d. Notepad
T.D.: Opnið Notepad. Notið „Edit“ og „Replace all“ til að skipta kommum út fyrir
punkta. Vistið skrá.
Smellið á IGOR á skjáborði
Data -> Load Waves -> Load general text
Veljið rétta möppu og skjal sem á að opna, Open -> Load: Windows -> New Graph
Veljið viðeigandi skrá undir Y wave(s). Notið “_calculated_” undir X Wave. Do it
Stækkið mynd
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Veljið hluta af mynd til að stækka frekar. Rammið inn svæði með bendli (vinstri
smellið), flytja bendil inn í ramma, vinstri smellið og veljið „Expand“.
Tíðnigildi toppa fundin: CTRL I eða Graph -> Show Info (krossbendlar birtast neðst)
Sækið krossbendil A (eða B) neðst í vinstra horni og dragið hann á einhvern toppinn.
Unnt er að fínstilla staðsetningu krossbendils með örvatökkum.
Lesið af x-ás gildi neðst (hér: 1482Hz) fyrir alla nothæfa toppa.
Merking ása: Graph -> Label Axis
Left -> (t.d.) útslag; Bottom -> t.d. Tíðni (Hz): Do it
Minnkið grafið til að taflan fyrir aftan birtist. Stimplið gildi toppa inn í töfluna
Windows -> New Graph. Veljið bylgju (wave) fyrir toppagildi (hér wave1): Do it
Punktar í stað línu á graf: Smellið á feril. Veljið “Markers” undir “Mode”.
Veljið t.d. hringi: Do it
Nálgið bestu beinu línu: Analysis -> Curve fitting
Veljið „line“ undir „Function“
Veljið viðeigandi bylgju (wave) undir Y Data (hér wave1): Do it
Halltala (b) (og skurðpunktgildi (a)) ásamt óvissugildum birtast í aðalglugga IGORS
(History, Hér: Untitled) oftast neðst til vinstri. Kannski þarf að færa til eða minnka
glugga
Further analysis relevant to measurements A and B:
Analysis:
By assuming the gases are ideal, find the heat capacity ratio in each case. Estimate the error in each of the
measurements and estimate the uncertainity in the value you obtain for the heat capacity ratio. Then, use the van
der Waals equation of state for N2 and CO2 and recalculate the heat capacity ratio (the van der Waals
coefficients can be found in chapter 1 of the text book by Silbey and Alberty*). How important is it to take nonideality into account in this case? Is the uncertainity in your determination of the heat capacity ration small
enough to make non-ideality detectable?
Proper theoretical treatment of the heat capacity of gases is now needed ( see for example reference 3*). It turns
out that vibrational motion often does not contribute fully to the heat capacity until the temperature is quite high.
This is because the quantization of vibrational energy gives large energy gaps between adjacent energy levels.
For this report, you first of all assume the classical result is right for all degrees of freedom, i.e. use equipartition
theorem. Recall that equipartition theorem says that each translational degree of freedom contributes kT/2 to the
internal energy (k is the Boltzmann constant), each rotational degree of freedom also contributes kT/2, but each
vibrational degree of freedom contributes kT (more than translation and rotational degrees of freedom because of
the potential energy increase as the bond lengths and bond angles are distorted). What is the predicted heat
capacity ratio for N2, Ar and CO2 using classical physics? Is it in agreement with your measured results
(carefully taking the uncertainty of the measurement into account)? At room temperature, the vibrational degrees
of freedom that have high frequency do not contribute the full kT to the internal energy, as you will see later
from quantum statistical mechanics. It is often a better approximation to skip completely the vibrational
contribution to the internal energy, rather than to include the classical, high temperature limit, kT. Repeat your
calculation of the heat capacity of N2 and CO2 now skipping the contribution from vibration. Does this give
better agreement with your measured values? Are they in agreement with your measurements, again taking into
account the experimental uncertainty? Finally, assume the contribution of translation and rotation is given
correctly by classical physics and evaluate, from the measured heat capacity ratio, the contribution of vibration
to the constant volume heat capacity. What fraction of the full, classical value does the vibrational contribution
turn out to be under the conditions of your experiment?
Derive an expression for the heat capacity ratio of a gas of bent molecules containing a total of N atoms, and of a
gas of linear molecules with N atoms. Could you use your measurements to tell whether CO 2 is linear or bent
(considering just the estimated uncertainity in the measurement, not the agreement between the predicted and
measured values)?
Perform calculations of the vibrational contribution to the heat capacity of CO2 and N2 using quantum statistical
mechanics (see for example www(3)* and (4)*) and compare with the experimentally estimated values obtained
from gamma.
Full report is required (see also HERE).
*
1) GNS7: "Experiments in Physical Chemistry" eftir C.W. Garland, J.W. Nibler og D.P.
Shoemaker, 7. útg., 2003, Experiment 13, page 199.
2) GNS8: "Experiments in Physical Chemistry" eftir C.W. Garland, J.W. Nibler og D.P.
Shoemaker, 8. útg., 2008.
3) "Physical Chemistry" eftir Robert J. Silbey & Robert A. Alberty, 4. útg., 2005.
4) www(1): http://www.chemistry.nmsu.edu/studntres/chem435/Lab3/
5) www(2): http://www.chemistry.mcmaster.ca/~ayers/chem2PA3/labs/2PA35.pdf
VEE-HC-12.doc /
VEE-HC-12.pdf
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