Experiment 3 Heat Capacity Ratio of Gases

Heat Capacity Ratio of Gases
Morgan Turano
mturano@chm.uri.edu
Office Hrs: Wed. 11 a.m., Pastore 201A

Purpose
To determine the heat capacity ratio for
monatomic and diatomic gases.
To understand and mathematically
model reversible & irreversible adiabatic
processes for ideal gases.
To practice to propagate error for
complex functions.

Key Physical Concepts
Heat capacity is the ratio of heat input/output to
temperature change in a substance
C = dq / dT
Where C = heat capacity, dq = change in heat,
dT = change in Temp.
And is defined as: γ = CP/ CV
Where:
CP = (∂H / ∂T)P
CV = (∂U / ∂T)V
Adiabatic processes occur when no heat is
exchanged between the system and the
surroundings (i.e. q=0)

Theory: Energy
Energy of gaseous molecules can be defined
as translational (ET), rotational (ER), or
vibrational (EV)
ETOT = ET + ER + EV
ET = (3RT)/2
ER = DRT/2 (D = 2 or 3)
EV = 3N – ET – ER
Where N = number of atoms, R = gas constant,
T = temperature, 3N = total degrees of freedom,
D = number of possible dimensional rotations

Theory: Heat Capacity
Heat Capacity (const. V): CV = (∂U / ∂T)V
Monatomic: CV = 3R/2
Diatomic: CV = 5R/2
Heat Capacity Ratio:
CP = CV + R
γ = CP / CV = 1 + (R/CV)
Reversible: γ = [ln(P1 / P2)] / [ln(P1 / P3)]
Irreversible: γ = [(P1 / P2) – 1] / [(P1 / P3) – 1]
The more molecular motions…
The greater the heat distribution
Therefore diatomic heat capacity > monatomic heat
capacity

Theory: Determination of
Heat Capacity Ratio
To determine heat capacity ratio, we will
subject a gas to an adiabatic expansion and
then allow the gas to return to its original
temperature via an isovolumetric process, during
which time it will cool.
In the above processes, three states of the gas
will be examined:
Before expansion: P1, V1, T1, n1
Immediately after expansion: P2, V2, T2, n1
After returning to room temperature: P3, V2, T1, n1
This expansion and warming can be modeled
two different ways.

Reversible Expansion
(Textbook)
Assume that pressure in carboy (P1) and
exterior pressure (P2) are always close enough
that entire process is always in equilibrium
Since system is in equilibrium, each step must
be reversible
Can generate smooth, well behaved analytical
functions from equation of state

Irreversible Expansion
(Lab Manual)
Assume that pressure in carboy (P1) and
exterior pressure (P2) are not close enough;
there is sudden deviation in pressure; the
system is not in equilibrium
Since system is not in equilibrium, it is not
possible to generate continuous functions
The equation of state is only valid at endpoints
and must be evaluated as such

Part I: Experimental Procedure
1) Set up apparatus, insert rubber stopper.
2) Open tubes A & B, attach C to manometer.
a) For Argon, attach A to Argon tank (input line).
Leave tube B open (output line).
b) For Nitrogen, attach B to Nitrogen tank (input line).
Leave tube A open (output line)
3) Allow gas to flow into carboy at 15 mbar for 15
minutes. Close the gas.
4) Close output line. Slowly open the gas a
small amount; hold down stopper.
5) When manometer reaches 60 mbar, turn off
gas. Manometer reading is now constant.
6) Record manometer reading (Man1).

Part I: Experimental Procedure
7) Remove stopper 2-3” and replace tightly as
quickly as possible. Hold down stopper.
8) Record manometer reading (Man2).
9) Record manometer reading again when
manometer reading is constant (Man3).
10) Repeat steps 2 - 9 for a total of six
measurements (three for Ar, three for N2).
-- Between runs of the same gas flush (step 3) for only 3 minutes.
Between different gasses flush for 15 minutes. Why?
11) Record lab temperature.
12) Record lab barometric pressure.

Part I: Experimental Procedure
Use
line A to
input Ar,
which reaches
the bottom of
the carboy.
Use
line B to
input N2,
which does
not reach the
bottom of the
carboy.
Set Up of Apparatus
Argon
manometer
http://itl.chem.ufl.edu/4411L_f00/gamma/eq14.gif

Part II: Data Analysis
P1 = Man1 + Barometer
P2 = Man2 + Barometer
P3 = Man3 + Barometer
Reversible: γ = [ln(P1 / P2)] / [ln(P1 / P3)]
Irreversible: γ = [(P1 / P2) – 1] / [(P1 / P3) – 1]

Part III: Laboratory Report
Title Page: Title, name, partner(s), date of
experiment.
Abstract: One (1) paragraph of what, why, how
and results.
Introduction: Discussion of purpose and
general nature of experiment, derive equations.
Theory: State all assumptions, define all
variables, give variations on formulas.
Discuss both reversible & irreversible adiabatic
expansion, and the difference between them.
Explain which you think closest resembles this
experiment.
Procedure & Original Data: Both signed.
Results: Data tabulated in order of calculations
that follow.

Part III: Laboratory Report
Calculations: (SHOW ALL WORK)
For each trial, determine γ for both reversible and
irreversible expressions.
Include at least one sample calculation of each type
of calculation used in numerical analysis.
Error Analysis: (SHOW ALL WORK)
Assume error of ±2 in the last recorded figure of
manometer and barometer.
Report γ as the average of all trials for each gas and
each expression.
Choose only one trial to propagate error in γ for each
expression; be sure to identify which trial is used.
Propagate errors in each pressure to obtain error in γ.
Compute expressions for statistical error in γ for both
reversible and irreversible processes.

Part III: Lab Report
Error Analysis (Cont’d)
Propagate error in each of the following:
P1 = Man1 + Barometer
P2 = Man2 + Barometer
P3 = Man3 + Barometer
Reversible: γ = [ln(P1 / P2)] / [ln(P1 / P3)]
Irreversible: γ = [(P1 / P2) – 1] / [(P1 / P3) – 1]

Part III: Laboratory Report
Summary of Data: Report all values with error
and determine (with math) the best value (rev. or
irrev.) of the average γ & determined error in γ.
Conclusions:
Discuss significance of results. Do they match
hypothesis/expectations?
Compare γ±ε(γ) for both reversible and irreversible
expansions. Do the two values lie within their
respective statistical errors?
Compare γ±ε(γ) with the expected values for a
monatomic and diatomic gas. Do the theoretical
results lie outside the experimental errors? If so,
discuss possible reasons for the lack of agreement.
Suggest how experiment might be improved.

Important Reminders
Use line A for Argon input
Use line B for N2 input
Make sure one partner is holding
stopper tightly in carboy from time = 0s.
When calculating pressures, add
manometer readings with barometric
pressure of room.
TA will only sign data AFTER you have
turned off the gases.
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