Heat Capacity Essay Research Paper Keith Griswold
Heat Capacity Essay, Research Paper
Heat Capacity Essay Research Paper Keith Griswold Essay Example
Keith Griswold March 13, 2000
Exp. Physical Chem.
Heat-Capacity Ratios for Gases
Cp = Cv + R
Cp = g
is related to the ability of the gas to make enlargement work. Heat capacity at changeless volume, Cv can be described utilizing the equipartition theory, which states that each manner of gesture will lend to a molecule or atom & # 8217 ; s energy.
E = E ( translational ) + E ( rotational ) + E ( vibrational )
Puting up a Cartesian co-ordinate system, translational gesture can happen in any of the three waies: ten, Y, or omega. Thus for a monoatomic gas energy can be represented as 3 ( RT/2 ) ; it is clear that no vibrational or rotational gestures contribute. Rotational gesture contributes to the energy of diatomic and polyatomic molecules ; they are easy accessible at room temperature therefore will significantly lend to g. Vibrations can be separated into two classs: bending and stretching, where the figure of manners can be described as 3N-5 for additive, and 3N-6 for nonlinear molecules. Vibrational degrees are non every bit accessible as rotational 1s are at room temperature, so it is valid to see them, at most, merely partly active ; the extent depends on certain belongingss of the molecule. Stretching manners tend to hold really high frequences giving manner to a little part to heat capacity ratios. It should be noted that electronic passages will be ignored since most molecules are in their electronic land province at room temperature. Using authoritative statistical mechanics to the equipartition theory, an look for the energy part of one mole of a gas from each manner of gesture is given as RT/2. Since heat capacity varies with temperature the undermentioned relationship is given:
Cv = ( dE/dT ) v. ( E is the internal energy )
Using these relationships, a theoretical value of g can be derived. In contrast, experimental values will be calculated through an adiabatic enlargement method initiated by Clement and Desormes. Discrepancies between the experimental values and those predicted by the equipartition theory will be examined.
Theoretical values for g can be calculated utilizing the equipartition theory and the relationship Cp = Cv + R. The gases of involvement for this experiment are argon, N, and C dioxide. Argon is monoatomic, therefore we will merely be looking at it & # 8217 ; s translational gestures. Cv will be 3R/2 and Cp will be 5R/2, go forthing g with a value of 1.6667. Nitrogen is a additive diatomic molecule, hence rotational manners must be accounted for ; a value of g will be calculated with and with out vibrational part. Cv for N without vibrational recognition is equal to 5R/2 and Cp is equal to 7R/2, go forthing g equal to 1.4000. Taking vibrational parts into history, Cv will be 3R and g will so be 1.3333. Carbon dioxide is triatomic, but is still additive so Cv is 5R/2 and Cp is 7R/2, allowing g equal 1.4000. With vibrational manners included, g peers 1.2222. All of these values are summarized in a tabular array below along with experimental 1s.
The adiabatic enlargement procedure will be carried out utilizing the experimental apparatus illustrated below.
A gas of involvement is placed in the carboy and force per unit area readings are so recorded. The carboy is closed off with a gum elastic stopper that is removed for a brief minute and so replaced. The gas in the carboy will momently make atmospheric force per unit area, and so shack to it & # 8217 ; s initial temperature, at which clip the force per unit area is recorded once more. It is necessary to enter the atmospheric force per unit area at the clip of the experiment. Three tests were run on each gas to obtain the undermentioned information. The force per unit area transducer used here was an unfastened tubing manometer incorporating dibutyl phthalate, so the force per unit area readings were converted to mmHg before used in computations.
Gas Trial P1 ( mmdi-but. ) P1 ( mmHg ) P3 ( mmdi-but. ) P3 ( mmHg )
Ar 1 155 773 60.0 766
2 145 773 60.0 766
3 240 780 50.0 765
N2 1 270 782 55.0 766
2 410 793 90.0 768
3 150 773 33.0 764
CO2 1 480 798 110 770
2 710 816 350 788
3 255 781 58.0 766
Barometric Pressure: 761.5 mmHg
Raw information was converted to mmHg by multiplying the recorded force per unit areas by ( 1.046g/cm3 ) / ( 13.55g/cm3 ) .
In order to cipher heat capacity ratios from the natural information it is necessary to handle this adiabatic enlargement as being reversible. Upon rapidly let go ofing the stopper, the upper and lower parts of gas signifier an fanciful surface between them, in which the lower part pushes reversible against the upper part. Work is done by the lower part of the gas forcing the upper part out of the ca
rboy. The relationship,
g = ln ( P1/ P2 ) /ln ( P1/ P3 )
can be derived where P2 is the barometric force per unit area at the clip of the experiment.
Gas Trial g g ( mean ) g g ( with vibrational parts
Ar 1 1.65
3 1.24 1.65 1.6667
N2 1 1.29
3 1.28 1.27 1.4000 1.3333
CO2 1 1.31
3 1.30 1.30 1.4000 1.2222
The entries that are non highlighted were left out of the mean since they lack preciseness. This disagreement will be attributed to uncertainnesss in the experimental process.
The experimental apparatus applied is questionable, but it does bring forth a sensible system for survey. The procedure is considered adiabatic because it is rapid and no appreciable sum of heat is transferred. It is besides non necessary to be concerned with molar concentrations since ratios are being measured. As would be expected, Ar is in most understanding with the equipartition theory ; little disagreements can be attributed to experimental uncertainnesss. A major ground for this understanding is the fact that argon issues as a monoatomic gas, where there are no vibrational or rotational manners. There is no concern that the low temperature employed would impede it & # 8217 ; s entree to allowed grades of freedom. Nitrogen, on the other manus is affected greatly by the fact that this experiment was conducted at room temperature. Rotation requires less energy, doing a nice degree of part from it, but vibrational manners are, at best, partly activated at 25 & # 176 ; C. There are a figure of factors that decide the of extent vibrational part to heat capacity ratios. Nitrogen gas consists of two N atoms connected by a ternary bond that bares a force invariable of 2,243 N/m. This ternary bond requires a batch of energy to hover and would necessitate highly high temperatures to hold complete activity from this vibrational manner. Since N gas is a diatomic molecule, it merely undergoes stretches that vibrate at low frequences, which contribute less to g at low temperatures. It seems that N is non a good campaigner for this experiment and behaves less ideally than Ar under the fortunes. An facet that is upseting is the fact that the experimental value for it & # 8217 ; s heat capacity falls lower that the theoretical one calculated that takes vibrational manners into history. Sing that there is sufficient grounds to propose that vibrational manners are non to the full activated, there has to be a ground for this unexpected consequence. The unexplained may merely lie wholly in the kingdom of experimental mistake, a fiddling account none the less. It could be possible that resonance varies the behaviour of the nitrogen/nitrogen bond doing it to exhibit deviated features. Carbon dioxide was the last gas to be studied ; it & # 8217 ; s triatomic with two dual bonds linking Os to the cardinal C. In a similar manner to nitrogen gas, C dioxide does non act harmonizing to the equipartition theorem at room temperature. Although the vibrational manners of CO2 are merely partly active, they contribute more that the 1s for N gas. This is a consequence of the bending capablenesss that CO2 has, which require less energy than stretches do. The bonds in C dioxide have force invariables of 1,857 N/m ; significantly less that the force invariable of N. Taking into history experimental uncertainnesss, the values calculated for CO2 from this process are what would be expected. The experimental value falls mediate both the theoretical 1s derived with and without vibrational parts, turn outing that for CO2, the vibrational manners are merely partly active at room temperature.
In analyzing C dioxide, we can see how structural one-dimensionality affects it & # 8217 ; s heat capacity ratio predicted by the equipartition theory. If it were non-linear, for illustration like SO2 or H2O, there would be a decreased figure of vibrational manners lending to it & # 8217 ; s energy. Accounting for the low vibrational gestures, g ( theoretical ) would increase somewhat to a value of 1.2500. It would be hard to decode between these two constructions based off experimental values of g, particularly with such a questionable experimental apparatus. It is necessary to recognize that the disagreement between the two ratios, posed by the difference in construction, is little ; such preciseness would be hard to accomplish.
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