Apparatus for Determining the Ratio of Specific Heats c_p/c_v

The tube. of oscillation, combined with an aslpirator (371 04), is used for determining the ratio cp/cv by the Rochardt method.

1. Description

A steel ball fits exactly into a 60 cm long precision glass tube having a diameter of about 16 mm. On delivery both ends of the tube are shut by rubber stoppers.

When the glass tube is held vertically and the lower end is closed with the finger before letting the ball fall into it, the latter will sink down very slowly. It takes considerable time to fall through the tube, for the air enclosed in the tube between finger and ball escapes very slowly through the narrow gap between ball and tube wall. When instead of the lower end, the upper end of the tube is closed with the finger, the ball sinks down just as slowly. This time the reason is that the air above the ball is rarefied, so that the atmospheric pressure is higher than the pressure in the enclosed volume and supports the ball. Due to the internal friction, the air can enter the tube only very slowly through the narrow gap between ball and tube wall.

When the hall is allowed to fall through the tube with both ends of the tube open, it falls down quickly. When the tube is then suddenly closed at one end, the ball is braked immediately and bounces up and down a few times before continuing' to sink down slowly.

2. Experiments

Before each experiment, carefully clean the tube by drawing a soft piece of leather or tissue through it on a thread.

Determination of the ratio -c_p/c_v

For the experiment, one needs an aspirator (371 04) which has a capacity of about 10 liters and has a rubber stopper with hole fitting the glass tube for the upper opening and a rubber stopper with hole and tap for the lower opening. The bottom of the aspirator is laid out with foam I rubber to moderate the impact of the ball, when this falls into the aspirator.

The precision glass tube is inserted vertically into the rubber bung with hole on top of the aspirator, and the lower tap is closed. If now the ball is allowed to fall into the glass tube, it performs harmonic oscillations on the air cushion formed by the enclosed volume of air. The oscillations are damped due to the unavoidable losses of energy by friction.

The letters mentioned below represent:

m = mass of the ball

A = cross-sectional area of the precision glass tube

V = volume of enclosed air

P_o = barometric pressure

p = pressure inside the bottle

g = acceleration due to gravity

c_p = specific heat at constant pressure

c_v = specific heat at constant volume

c = c_p/c_v

The ball is in equilibrium if the pressure p inside the aspirator is equal to the sum of the atmospheric pressure P_o and the pressure due to the weight of the ball:

(1) .

When the ball moves a distance x beyond its equilibrium position, the pressure changes by dp. By this a force A dp is exerted on the ball, imparting to it an acceleration . Then by Newton's second axiom:

(2) .

This process may be considered practically adiabatic. Therefore:

(3) .

By differentiation:

(4)

(5) .

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

(6) .

By substituting (6) in (5):

(7)

By substituting (7) in (2):

(8) .

This is the differential equation of a harmonic oscillation from which the angular frequency can-be deduced, which is:

(9)

From this the period of the harmonic oscillation performed by the ball is found:

(10)

From this follows for c = c_p/c_{v :}

(11) .

As all quantities on the right side of equation (11) are accessible to measurement, x can be determined in this way.

The accuracy of the result depends above all on the exact determination of period T whose square enters into equation (11). Therefore, the time required for ten oscillations, is measured best several times (ten oscillations are still conveniently observed) and their arithmetic mean value T is thus formed. This is then used for the calculations.

Internal friction of gases

If a glass tube is evacuated, the number of existing gas molecules will decrease, whereas the free path of residual molecules will increase. This causes the steel ball falling within an evacuated glass tube to experience in the beginning a small gain of speed while the pressure is decreasing. The more the free path of gas molecules increases, the smaller will be the internal friction of gases ~ which somewhat increases the sinking speed of the steel ball. If, however, the free path of gas molecules gets into the interval between steel ball and glass wall, the friction between ball and glass wall will be greatly reduced. If this pressure is attained, the falling speed of the ball suddenly rises (Fig. 2).

For this experiment, one uses the tube of oscillation without aspirator. One end of the tube is closed by a rubber stopper. The other end is connected to a vacuum pump, e. g. a two-stage gas ballast pump S 1,5 (101 01S)i, via a vacuum tubing

(172 02) and the adapter made of glass (554 00) or the adapter made of metal (378 00). For the measurement of the pressure, the Bourdon gauge (379 15) is fitted on top of the adapter's male ground joint.

During evacuation, the end of the glass tube closed by the stopper, as wellas the ball, must be at the bottom. If the desired pressure has been attained, the falling speed of the ball con be measured after the tap has been shut and the tube turned round.

Notes

1. The five-digit numbrs quoted in parentheses refer to the catalogue numbers of the respective apparatus.

2. The specifications and illustratlons are nol binding in avery detail for the deslgn of the apparatus, it is our policy always to keep our manufacturing programme right up to date so that it makes full allowance for the latest knowledge acquired in all scientific and ledlnical fields.