Heat Capacity Essay Research Paper Keith Griswold

Heat Capacity Essay, Research Paper

Keith Griswold March 13, 2000

Exp. Physical Chem.

Dr. Humphrey

Heat-Capacity Ratios for Gases

Cp = Cv + R

The ratio,

Cp = g

Cv

is related to the ability of the gas to do expansion work. Heat capacity at constant volume, Cv can be described using the equipartition theory, which states that each mode of motion will contribute to a molecule or atom’s energy.

E = E(translational) + E(rotational) + E(vibrational)

Setting up a Cartesian coordinate system, translational motion can occur in any of the three directions: x, y, or z. Thus for a monatomic gas energy can be represented as 3(RT/2); it is clear that no vibrational or rotational motions contribute. Rotational motion contributes to the energy of diatomic and polyatomic molecules; they are easily accessible at room temperature therefore will significantly contribute to g. Vibrations can be separated into two categories: bending and stretching, where the number of modes can be described as 3N-5 for linear, and 3N-6 for nonlinear molecules. Vibrational levels are not as accessible as rotational ones are at room temperature, so it is valid to consider them, at most, only partially active; the extent depends on certain properties of the molecule. Stretching modes tend to have very high frequencies giving way to a small contribution to heat capacity ratios. It should be noted that electronic transitions will be ignored since most molecules are in their electronic ground state at room temperature. Applying classic statistical mechanics to the equipartition theory, an expression for the energy contribution of one mole of a gas from each mode of motion is given as RT/2. Since heat capacity varies with temperature the following 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 expansion method initiated by Clement and Desormes. Discrepancies between the experimental values and those predicted by the equipartition theory will be examined.

Theoretical Calculations:

Theoretical values for g can be calculated using the equipartition theory and the relationship Cp = Cv + R. The gases of interest for this experiment are argon, nitrogen, and carbon dioxide. Argon is monatomic, therefore we will only be looking at it’s translational motions. Cv will equal 3R/2 and Cp will equal 5R/2, leaving g with a value of 1.6667. Nitrogen is a linear diatomic molecule, therefore rotational modes must be accounted for; a value of g will be calculated with and with out vibrational contribution. Cv for nitrogen without vibrational acknowledgement is equal to 5R/2 and Cp is equal to 7R/2, leaving g equal to 1.4000. Taking vibrational contributions into account, Cv will equal 3R and g will then be 1.3333. Carbon dioxide is triatomic, but is still linear so Cv is 5R/2 and Cp is 7R/2, letting g equal 1.4000. With vibrational modes included, g equals 1.2222. All of these values are summarized in a table below along with experimental ones.

Experimental:

The adiabatic expansion process will be carried out using the experimental setup illustrated below.

Apparatus Scheme

A gas of interest is placed in the carboy and pressure readings are then recorded. The carboy is closed off with a rubber stopper that is removed for a brief moment and then replaced. The gas in the carboy will momentarily reach atmospheric pressure, and then reside to it’s initial temperature, at which time the pressure is recorded again. It is necessary to record the atmospheric pressure at the time of the experiment. Three trials were run on each gas to obtain the following data. The pressure transducer used here was an open tube manometer containing dibutyl phthalate, so the pressure readings were converted to mmHg before used in calculations.

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 data was converted to mmHg by multiplying the recorded pressures by (1.046g/cm3)/(13.55g/cm3).

Results:

In order to calculate heat capacity ratios from the raw data it is necessary to treat this adiabatic expansion as being reversible. Upon quickly releasing the stopper, the upper and lower portions of gas form an imaginary surface between them, in which the lower portion pushes reversible against the upper portion. Work is done by the lower portion of the gas pushing the upper portion out of the carboy. The relationship,

g = ln(P1/ P2)/ln(P1/ P3)

can be derived where P2 is the barometric pressure at the time of the experiment.

Experimental Theoretical

Gas Trial g g(average) g g(with vibrational contributions

Ar 1 1.65

2 1.65

3 1.24 1.65 1.6667

N2 1 1.29

2 1.27

3 1.28 1.27 1.4000 1.3333

CO2 1 1.31

2 1.98

3 1.30 1.30 1.4000 1.2222

The entries that are not highl

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