# Determination of the Adiabatic Constant

## Contents

# Description of the Experiment

The purpose of this experiment is the determination of the ratio between the specific heat of air (constant pressure and constant volume), through the use of adiabatic oscillations of an embolus of known dimensions.

**Links**

- Video: rtsp://elabmc.ist.utl.pt:554/gamma.sdp
- Laboratory: Advanced in e-lab.ist.eu[1]
- Control room: Cp/Cv
- Level: ****

# Experimental Apparatus

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

# Protocol

Ruchhardt’s method (see bellow) is a way to determine the specific heat of a gas in a very precise way, but it is very sensitive to the measurement of the oscillations period. Because of this, extra care in this measurement is recommended and 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 judiciously, exploring all the information that it can give. After a reference volume is selected, the embolus is agitated so that it oscillates freely around the 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, allowing the parameter \( \gamma \) to be free as well as the volume and pressure, the measure 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 the diameter have a 0.5% precision.

# Data Analysis

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

# Theoretical Principles

With this method, it is possible to determine the ration between the specific heat of a gas through experimentation. If the gas in study is the atmospheric air (mostly diatomic), this ratio should be 1.4.

**Ruchhardt's Method**

If we consider a piston without friction, oscillating freely in a cylinder of volume \( V_0 \), with pressure \( p \), then the force exerted upon the piston ( \( m \ddot{y} \) ) equals the force of gravity minus the variation of pressure upon the piston( \( A \Delta p \) ).

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

The variation of pressure for small oscillations in volume is:

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

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

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

From the above equation we have:

[math] \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} [/math]

and

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

simplifying

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

We make

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

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

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

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

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

where \( r \) is the cylinder radius. A more precise estimation can be achieved using the differential equation considering the dumping effect caused by friction. Is such a situation you could consider friction being proportional to velocity leading to:

```
[math]
\ddot{y} + 2\lambda\omega \dot{y}+\omega ^2 y + g = 0
[/math]
```

Considering again the change in the origin, the result of such an equation leads to:

```
[math]
y' = y'_{0} e^{-\lambda \omega t}cos( \sqrt{1 - \lambda^2}\omega t + \phi)
[/math]
```

where the period leads to a slight correction due to the dumping factor.