3.2. ADIABATIC COEFFICIENT 91

pVκ= const

∆pF = A

Rubberbung

∆p

Differentialmanometer

Ball

D

pumpHand

GasMarriot’s

flask

Abbildung 3.1: Scheme of the Ruchardt method for the ratio cpcv

measurement

3.2 Adiabatic Coefficient

Purposes

In this experiment the ratio cpcv

is measured by the Ruchardt method.

Keywords

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 Ruchardt method

The Ruchardt method to determine the ratio cpcv

for a gas is based on the measurement ofoscillations 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 tubeis inserted vertically into a glass bottle (Marriot’s flask) which has a rubber stopper with a holefitting 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 raisesthe ball along the tube. After air influx is stopped the ball will sink down very slowly. It takesconsiderable time to fall through the tube, for the air enclosed in the tube below the ball escapesvery 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 startsto oscillate up and down a few times before continuing to sink down slowly. The oscillations aredamped due to the unavoidable losses of energy by friction. Variations of the air pressure insidethe bottle during these oscillations have to be measured and stored. Analysis of these data inaccordance 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 = πD2/4 - 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 = cpcv

- adiabatic coefficient

The ball is in equilibrium if the pressure p inside the bottle is equal to the sum of the atmosphericpressure p0 and the pressure due to the weight of the ball:

p = p0 +mg

A(3.1)

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 velocityFfr = −αfr dxdt . Then by Newton’s second law:

md2x

dt2= Adp− αfr

dx

dt(3.2)

The process may be considered practically adiabatic. Therefore, according to 3.15 (Section 3.1.1)

pV κ = const (3.3)

By differentiation:V κdp+ pκV κ−1dV = 0 (3.4)

3.2. ADIABATIC COEFFICIENT 93

dp =pκ

VdV (3.5)

The ball was supposed to move a distance x in the glass tube; this gives a change of volume

dV = Ax (3.6)

By substituting 3.6 in 3.5

dp = −pκAxV

(3.7)

The equation of motion 3.2 takes now the form

d2x

dt2− αfr

dx

dt+pκA2

mVx = 0 (3.8)

This is the differential equation of harmonic oscillations with damping from which the angularfrequency of the ball oscillations can be deduced, which is

ω =

√pκA2

mV(3.9)

from this follows for κ = cpcv

:

κ = ω2mV

pA2(3.10)

All the quantities on the right side of equation 3.10 are accessible to measurements, therefore κcan 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 hPa to bemeasured. It can be connected directly to the Sensor-CASSY.

Technical data:

94

• Measuring ranges: ±0.7 hPa,±7 hPa,±70 hPa

• 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 tubeshould 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 thestand.

• Connect the outlet of the Marriot’s flask with the hand pump and with the differentialpressure 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 cpcv

96

The measurement of the parameter V can be performed by weighing of the empty bottle and ofthe bottle filled with water. This can be done only after performing the oscillation measurementbecause 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 mea-sured for both Marriot’s flasks.All the measurements have to be done several times. Evaluation of sources and values of syste-matic 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 calculatedusing all the measurements done. The resulting κ value measured in the experiment have to becalculated 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 theexperiment performance, etc.) has to be presented.