3.2 Adiabatic Coefficient

3.2. ADIABATIC COEFFICIENT 91
Rubber
bung
Marriot’s
flask
Ball
F = AΔp
D
Gas
pV κ = const
Differential
manometer
Δp
Hand
pump
Abbildung 3.1: Scheme of the Rüchardt method for the ratio cp
cv measurement
3.2 Adiabatic Coefficient
Purposes
In this experiment the ratio cp
Keywords
cv
is measured by the Rüchardt method.
Specific heat, equation of state, processes of state transformation (adiabatic, isothermic, isobaric),
pressure oscillations
Literature
See list of sources in Section 3.1
3.2.1 Rüchardt method
The Rüchardt method to determine the ratio cp
for a gas is based on the measurement of
cv
oscillations of a steel ball supported by a column of the gas in a precision glass tube.
Scheme of the experiment is shown in Fig. 3.1.

92
The ball fits exactly into the tube having a diameter of about 16 mm. The precision glass tube
is inserted vertically into a glass bottle (Marriot’s flask) which has a rubber stopper with a hole
fitting the glass tube. Gas pressure inside the bottle is measured by a differential manometer.
A hand pump can provide a slow influx of air into the bottle. The increased air pressure raises
the ball along the tube. After air influx is stopped the ball will sink down very slowly. It takes
considerable time to fall through the tube, for the air enclosed in the tube below the ball escapes
very slowly through the narrow gap between ball and tube walls.
If the ball hight is increased sharply by the use of two small permanent magnets the ball starts
to oscillate up and down a few times before continuing to sink down slowly. The oscillations are
damped due to the unavoidable losses of energy by friction. Variations of the air pressure inside
the bottle during these oscillations have to be measured and stored. Analysis of these data in
accordance with a theory described below allows to determine the air adiabatic coefficient κ.
The parameters relevant for the analysis are
• m - mass of the ball
• A = πD 2 /4 - cross-section area of the glass tube
• V - volume of enclosed air
• p0 - atmospheric pressure
• p - pressure inside the bottle
• cp - specific heat at constant pressure
• cv - specific heat at constant volume
• κ = cp
cv
- adiabatic coefficient
The ball is in equilibrium if the pressure p inside the bottle is equal to the sum of the atmospheric
pressure p0 and the pressure due to the weight of the ball:
p = p0 + mg
A
When the ball moves a distance x beyond its equilibrium position, the pressure changes by dp.
By this a force Adp is exerted on the ball. Friction force is proportional to the ball velocity
. Then by Newton’s second law:
Ffr = −αfr dx
dt
m d2x dx
= Adp − αfr
dt2 dt
The process may be considered practically adiabatic. Therefore, according to 3.15 (Section 3.1.1)
By differentiation:
(3.1)
(3.2)
pV κ = const (3.3)
V κ dp + pκV κ−1 dV = 0 (3.4)

3.2. ADIABATIC COEFFICIENT 93
dp = pκ
dV (3.5)
V
The ball was supposed to move a distance x in the glass tube; this gives a change of volume
By substituting 3.6 in 3.5
dp = − pκAx
V
The equation of motion 3.2 takes now the form
dV = Ax (3.6)
d2x dx pκA2
− αfr + x = 0
dt2 dt mV
(3.8)
This is the differential equation of harmonic oscillations with damping from which the angular
frequency of the ball oscillations can be deduced, which is
ω =
pκA2 mV
(3.9)
from this follows for κ = cp
cv :
2 mV
κ = ω
pA2 (3.10)
All the quantities on the right side of equation 3.10 are accessible to measurements, therefore κ
can be determined in this way.
3.2.2 Setup and Carrying out the Experiment
Apparatus
Oscillation tube;
Stand base, V-shape;
Stand rod, 50 cm;
Clamp with jaw clamp;
Two Marriot’s flasks of different volumes;
Hand vacuum and pressure pump;
Differential pressure sensor;
Absolute pressure sensor;
Sensor-CASSY;
PVC tubes;
Connectors;
Permanent magnets;
Digital balance;
Precision micrometer.
The pressure sensor enables differential pressures ∆p = p1 − p2 between 0 and ±70 hP a to be
measured. It can be connected directly to the Sensor-CASSY.
Technical data:
(3.7)

94
• Measuring ranges: ±0.7 hP a, ±7 hP a, ±70 hP a
• Resolution: 0.05% of measuring range
The technical data of the absolute pressure sensor are described in Section 3.1.2.
Setup
Set up the experiment as shown in Fig.3.2:
• Insert the oscillation tube vertically into the dry Marriot’s flask. Both sides of the tube
should be closed to avoid fast movements of the ball inside the tube.
• Fix the vertical position of the tube by clamping it with the jaw clamp connected with the
stand.
• Connect the outlet of the Marriot’s flask with the hand pump and with the differential
pressure sensor using the PVC tubes and T-connector.
• Insert the pressure sensor in the INPUT A channel of the CASSY box.
• Open the upper end of the oscillation tube.
The setup is ready for the oscillation measurements.
Carrying out the experiment
The parameters which define the value of the adiabatic coefficient κ, namely
• ω - the ball oscillation frequency
• A = πD2
4 - cross-section of the tube
• m - mass of the ball
• V - volume of the bottle
• p - gas pressure in the equilibrium state of the ball
have to be measured using the provided tools:
• relative pressure sensor
• absolute pressure sensor
• precise micrometer
• digital balance

3.2. ADIABATIC COEFFICIENT 95
Abbildung 3.2: Experiment setup for measuring the ratio cp
cv

96
The measurement of the parameter V can be performed by weighing of the empty bottle and of
the bottle filled with water. This can be done only after performing the oscillation measurement
because the air inside the Marriot’s flask has to be dry then.
Ball oscillation frequencies for different ball positions (heights) inside the tube have to be measured
for both Marriot’s flasks.
All the measurements have to be done several times. Evaluation of sources and values of systematic
and stochastic errors for all the measured parameters should be presented.
The values of the adiabatic coefficient κ together with its measurement errors should be calculated
using all the measurements done. The resulting κ value measured in the experiment have to be
calculated using either weighted mean or linear regression method (or both).
Analysis of the κ error budget (sources of the dominant errors, possible ways to improve the
experiment performance, etc.) has to be presented.