Memorial University of Newfoundland Department of Physics and Physical Oceanography Physics 2053 Laboratory Isothermal and Adiabatic Processes

Introduction

This experiment measures the pressure, temperature and volume changes in air as it is slowly compressed or expanded under isothermal conditions, or rapidly under near adiabatic conditions.

Theory

The Ideal Gas Law can be written as

pV = nRT (1)

so that the pressure is inversely proportional to volume when held at constant temperature (isothermal).

An adiabatic process takes place when no thermal energy enters or leaves the system. This occurs if the system is perfectly insulated or if the process occurs so rapidly that there is no heat transfer. The first law of thermodynamics for an adiabatic process can be stated as:

dQ = nCvdT + pdV = 0 (2)

where Cv is the molar specific heat at constant volume. If we write the ideal gas law as

P dV + V dp = nRdT

and solve for dT : P dV V dp dT = + (3) nR nR

1 and substitute Eq (3) into Eq (2), we obtain

dQ = CpP dV + CvV dP

where Cp is the molar specific heat at constant pressure. Since dQ = 0 we obtain

C P dV dP p + = 0 CvPV P or dV dP γ + = 0 V P or γ ln V + ln P = const or PV γ = k (4)

This is the standard form of the adiabatic gas law.

Apparatus

A piston of diameter 4.448 cm moves inside a cylinder which has pressure and temperature sensors mounted inside. Each sensor produces a small voltage which is recorded by the computer.

Setup and Procedure

1. Connect the volume cable to Channel A of the Pasco interface box and the temperature and pressure sensors to Channels B and C respectively.

2. Start the DataStudio program and click on Create Experiment. A picture of the Pasco box will be displayed, with each channel circled in yellow.

3. Click on Channel A and select Voltage Sensor. Then click OK. Set the sample rate to 2 seconds; under Measurements, select Voltage (ChA). Repeat for the other two

2 sensors, selecting Voltage Sensor also. The picture on the computer screen will have three icons attached to it, corresponding to the three sensors.

4. To display the voltage output from each sensor, double click on the Digits icon (under the Display heading on the left hand side of the screen). Then select Voltage from the ‘Choose a Data Source’ screen (this will be either ChA, ChB or ChC). Repeat for the other sensors. Click ‘Start’ so that the voltages from all three sensors are displayed. Click on Stop to stop the data accumulation.

5. To two sets of apparatus in the lab are identified by serial numbers 2048 and 2223. For

both models the pressure in kilo Pascals is 100 times the output voltage, VC . Volume and temperature depend on the model used:

• Measurement of Volume: You can assume that the relationship between vol- ume in cm3 and voltage follows a straight line relation of the form

Volume = 34.32 × VA + 84.98 (2048)

= 32.70 × VA + 86.40 (2223)

• Measurement of Temperature: Temperature is obtained from voltage using the equation,

T (K) = 38.86VB + 262.9 (2048)

= 45.51VB + 256.6 (2223)

Isothermal Compression and Expansion

In this part of the experiment you will observe the pressure and volume changes which occur as the temperature is kept constant.

1. Double click on the Graph icon and select Pressure as the data source (i.e., the voltage output from the pressure sensor).

3 2. A set of axes will appear, with voltage on the y- axis and time on the x-axis. Click the ‘Time’ label and replace it with the voltage corresponding to the volume.

3. Click Start. Very Slowly move the piston up and down in small increments using the handle. [It is suggested that you take at least ten minutes to move the piston over the whole distance.] Note the starting temperature (voltage) and try not to let this vary by more than about 0.02 volts each time you move the piston. Click Stop when you have a complete set of data.

4. Export your data to the computer desktop and plot a suitable graph to show that pressure is inversely proportional to volume (i.e., use a power law fit or log-log plot to show that P ∝ 1/V ). There is no need to use the conversion factors because pressure and volume are both proportional to voltage.

Adiabatic Compression

1. Reset the sample rate to 100 Hz. Click Start and quickly compress the gas. Then click Stop. Import the raw data into your graphing program.

2. Create a data file which contains pressure, temperature and volume as a function of time. Keep only the data points corresponding to the time period when these quantities are changing. Plot a graph to show how these quantities vary with time. It should look something like Figure (1).

3. Plot a graph of pressure versus volume for the adiabatic compression. The form of the curve will be very similar to the isothermal case, but because pV γ = const, a plot of log(p) versus logV will give a straight line of slope −γ. Determine γ from your data.

[Optional] Work done in compressing the Gas

We can use Eq (1)to obtain a second form of the law as

(γ−1) (γ−1) T1V1 = T2V2 (5)

4 Figure 1: Typical results from adiabatic compression

Rearranging Eq (4) gives P = k/V γ and hence the work required to compress the gas is:

V V Z Z V2 dV V (1−γ)  2 V (1−γ)  2 W = P dv = k = k = (P V γ) V γ 1 − γ 1 1 1 − γ V1 V1 V1

Therefore, P V γ   W = 1 1 V (1−γ) − V (1−γ) (6) 1 − γ 2 1

1. Use your starting values of pressure and temperature, and equations (4) and (5) to cal- culate the final temperature and pressure predicted by the adiabatic gas law. Compare these with the values obtained experimentally.

2. Use Eq (6) to calculate the work done on the gas during the adiabatic process, and com- pare your result with the answer you obtain using the numerical integration function to calculate the area under your adiabatic p − V curve.

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