Determination of the Adiabatic Constant

Description of the Experiment

This experiment allows for the determination of the ratio between air's specific heats (constant pressure and constant volume), through the use of adiabatic osculations of an embolus of known dimensions.

Experimental Apparatus

The apparatus is composed of a syringe, the embolus with a weight of 26.4 gram and 18.9mm diameter. The embolus has reduced friction due to graphite lubrication and the fact that the apparatus is in the vertical position.

Protocol

Ruchhardt’s (see bellow) method is a way to determine the specific heats of a gas in very precise way, but is very sensitive to the measure of the osculation period. For that reason, extra care in this measure is recommended. Thus, two methods are used to determine this quantity: the waveform recorded by the pressure transducer and the average period, digitally determined. The data must be used wisely, exploring all the information that they can give. After a reference volume is selected, the embolus is disturbed so that it oscillates freely around it's equilibrium position. \( \gamma \) can be inferred from the oscillation period.

Advanced Protocol

By redoing the experiment for several volumes, a better adjustment can be achieved between the experimental data and the theoretical function. When adjusting the experimental data, letting not just the \( \gamma \) parameter free but also the volume and pressure, the measure's precision can be increased, since atmospheric pressure can have variations of up to 1% and because the volume measured will have a systematic error due to the various external connections to the syringe. It should be noted that the piston mass and diameter have a 0.5% precision.

Data Analysis

By using Fitteia, one can plot the experimental results and adjust a theoretical function with certain parameters. This file is an example of a fit for this experiment (right-click on the link and "Save As").

Theoretical Principles

Ruchhardt's Method

With this method, it is possible to determine the ration between the specific heats of a gas through experimentation. If the gas in study is the atmosferic air (mostly diatomic), the ratio is 1.4. If we consider an embolus without friction, oscillating freely in a cylinder of volume \( V_0 \), with pressure \( p \), then the force exerted upon the embolus ( \( m \ddot{y} \) ) equals the force of gravity minus the variation of pressure upon the embolus ( \( A \Delta p \) ).

\[ -mg+A \Delta p = m \ddot{y} \]

The variation of pressure for small oscilations in volume is:

\[ \Delta p = \frac{\partial p}{\partial V} | _{V = V_0} \]

if we consider a fast enough process so that no exchange is heat occurs (adiabatic process)

\[ pV^{\gamma} = p_0 V_0 ^{\gamma}, \quad p = \frac{ p_0 V_0 ^{\gamma} }{ V^{\gamma} } \]

From the above equation we have:

\[ \frac{\partial p}{\partial V} | _{V = V_0} = - \gamma frac{ p_0 V_0 ^{\gamma} }{ V^{\gamma +1} } | _{V = V_0} = - \gamma \frac{p_0}{V_0} \]

e

\[ -mg+ A (- \gamma \frac{p_0}{V_0} \Delta V) = m \ddot{y} , \text{ where } \Delta V = Ay \]

simplifying

\[ \ddot{y} + \gamma \frac{p_0 A^2}{m V_0} y+g = 0 \]

We make

\[ \gamma \frac{p_0 A^2}{m V_0} = \omega ^2, \text{ so that } \ddot{y} + \omega ^2 y + g = 0 \]

Changing the point of origin to the embolus's equilibrium position, we can easily see that this is the equation for the motion of a frictionless harmonic oscillator

\[ \ddot{y}' + \omega ^2 y' = 0 \text{ com } y = y' - \frac{g}{\omega ^2} \text{ and } \omega ^2 = (\frac{2 \pi}{T})^2 = \gamma \frac{p_0 A^2}{m V_0} \]

Measuring the oscilation period, \( T \), we can determine \( \gamma \)

\[ \gamma = \frac{4mV_0}{p_0 r^4 T^2} \]

where \( r \) is the cilinder radius.