Where h is specific enthalpy,
T is thermodynamic temperature,
s is specific entropy.
and since specific enthalpy h is given by:
Eq. 2
Where e is specific energy,
P is absolute pressure,
v is specific volume
the specific Gibb’s function can be re-written:
. Eq. 3
For infinitely incremental increases in a closed simple fluid system:
and
which gives . Eq. 4
Also since ,
it follows that . Eq. 5
In a two-phase fluid system, for equilibrium between liquid and vapour phases:
So, from Eq. 1:
Re-arranging:
Eq. 6
For in infinitely incremental increase in pressure, dP:
=> Eq. 7
Integration of the Clausius-Clapeyron Equation [1]
Since ,
If it is assumed that this volume of gas obeys the ideal gas law : Eq. 8
then .
Equation 7 above can then be simplified to:
Eq. 9
Integrating both sides
If it is assumed that hfg is constant,
then
which becomes:
=> Eq. 10
Apparatus
Fig. 1
Fig. 2
Experimental Procedure
-
Pressures above Atmospheric Pressure
The equipment in figure 1 above was used for this part of the experiment. The water in the sealed pressure chamber was brought to boiling point, and the safety valve was opened so that air could escape. This was done as quickly as possible in order that the amount of water lost as vapour (steam) could be minimised. Once the water had reached boiling point, the rate of heating was reduced until a steady reading on the thermometer was observed.
At this point the safety valve was closed and the rate of heating was increased to increase the pressure of the system. The heating was continually adjusted until the pressure gauge and thermometers showed steady readings, which were then recorded. It was my job to read the thermometer and announce the reading to the rest of the group so that they could record it.
Heating was then removed and the pressure of the system was allowed to fall to the next lowest reading. As before the rate of heating of the system was manipulated so that the pressure and temperature of the system were steady and these values were recorded.
This process was repeated and observations recorded for a number of different pressures until the pressure was approximately atmospheric. At this point the experiment was halted and the equipment was made safe.
-
Pressures below Atmospheric Pressure
The second set of equipment, shown in figure 2, was used for this part of the experiment. The valves in the vacuum pipe were opened and the voltage across the heating element was varied with a potentiometer to bring the water in the beaker to boiling point. Once the water had begun to boil, the voltage was lowered so that the water boiled more gently. The thermometer was observed until the temperature reached a steady value, and this was recorded.
At this point the water valve was opened partially, and the vacuum valve was opened and continually adjusted until the manometer showed a steady reading at the first pressure to be observed. Then the thermometer was observed until a steady reading was reached and this was recorded against the pressure in the chamber. As in the experiment above, it was my job to read the thermometer on behalf of the group since there wasn’t enough space and time to allow everyone to use the equipment individually.
The vacuum valve was then opened more fully to allow the pressure to fall to the next lowest value to be recorded, and as above adjusted until the manometer showed a steady reading, and then the steady reading on the thermometer was recorded. This was repeated for a range of pressures up to the maximum pressure difference on the manometer’s scale. The experiment was then halted again and the equipment was made safe.
Results
These were the recordings made during the experiment.
Pressures above Atmospheric
Fig. 3
(Temperatures recorded on the Celsius scale were converted to Kelvin by adding 273.2 K)
Pressures below Atmospheric
Fig. 4
(Pressures recorded in mmHg were converted to kPa by multiplying by 0.1333, and temperatures recorded on the Celsius scale were converted to Kelvin by adding 273.2 K)
Atmospheric Conditions
Fig. 5
These results are displayed on the figures below.
Fig. 6
Fig. 7
Discussion
While the observations were being recorded it quickly became obvious that there was an approximately linear relationship between temperature and pressure. This can be seen most clearly on the graph (figure 6) for the high pressure experiment. For the low pressure experiment the relationship appears to be approximately linear, but as the pressure approaches atmospheric, the relationship becomes more polynomial and the temperature appears to be more closely dependant on the absolute pressure.
These observations are supported by the results pictured in the logarithmic graph (figure 7). As might be expected from Eq. 10, both of the lines have a negative gradient. It can still be seen however that, especially for the low pressure experiment, the relationship between the two variables is approximately but not exactly linear. The results from this experiment appear to show that the linear relationship is lost as the temperature increases.
There are a number of factors that could start to explain this phenomenon.
Firstly, experimental error needs to be taken in to account. It is important to recognise the limitations of the laboratory equipment that was used in the experiment. For example it would not be possible to set up a perfect vacuum in the second set of equipment, nor to be certain that all of the air had been expelled from the chamber in either experiment and that the substance being observed was pure water. It would not be wise to assume that in either experiment it was possible to maintain a standard heating or cooling rate. Also the equations do not take in to account the fact that the substance being observed may have been able to do work on the surroundings by heating – neither set of equipment was perfectly insulated and indeed the chambers did seem to be radiating heat.
In the low pressure experiment particularly, it was possible for the vapour to escape the equipment which would have a significant effect on the shape of the results. In my opinion this was the most important factor in causing the results to appear more curved than linear, further supported by the fact that in the high pressure experiment (where the amount of substance being tested did not change throughout the experiment) the results form an almost perfect straight line. It is reasonable to assume that since there were fewer water particles in the chamber at the end of the experiment than at the beginning, the pressure would be lower than expected and this is exactly what has been observed from the results.
It should be noted that the results will be affected also by human error. The fact that readings were taken manually could have a bearing on their accuracy – for example the thermometer readings can only be accurate to 1 degree on the Celsius scale since that was the smallest division on the scale. Although the group strived to be consistent, parallax errors can occur when scales are read from different angles. This might have been avoided if digital or automatic equipment had been used to take the readings, which would also have been able to take readings at more frequent intervals giving more results from which to draw conclusions.
The dashed line in the figure below shows what might have been observed had the substance been completely trapped in the equipment, and to what degree the results may be inaccurate.
Fig. 8 – Estimate of Experimental Error
The maximum deflection of the curve from the expected straight line of a perfect relationship can be seen above, where in the low pressure experiment the pressure would have been observed to be 100kPa at a temperature of 372 K when it might have been reasonable to expect it at a temperature of 390 K.
This represents a temperature difference of 18K. Since the temperature was expected to be 390 K, and that analyzing the results this was the largest deflection from the perfect relationship it would be reasonable to assume that the maximum error in the observations is
18/390 *100 = 4.6% . This would need to be increased if human error is also going to be taken in to account.
The derivation of equation 10 relied on some assumptions, namely that the volume of the fluid is negligible in comparison to the volume of the vapour, that the value of hfg can be considered constant and that water vapour is an ideal gas.
The first is a valid assumption since the volume of a gas is so much greater than the volume of a liquid.
So if the volume of the liquid were taken in to account separately, it would be such a tiny factor that it would hardly affect the result of the calculation. In fact it has been shown that it would lead to an error of 0.1% so instead it should be included in the +constant factor of the equation.
Secondly that hfg is constant. It has been shown that [4] “the heat absorbed by a system at constant pressure is equal to the change in its enthalpy”. So enthalpy is actually dependant on temperature. However in this experiment pressure was relatively low, and for low pressures hfg is approximately constant [3]. Also we kept temperature and pressure constant for each recording. Since the temperature was constant the system was not absorbing heat (for the instant of the recording) so it is reasonable to assume that hfg is constant.
Finally for real gases at low pressure is almost constant. Since an ideal gas is one where and R is a constant, a real gas at low pressure can be approximated as an ideal gas [6]. Since the pressures in this experiment weren’t extremely high, this is a reasonable assumption.
Comparison to Accurately Measured Values
In the integrated form Clausius-Clapeyron Equation is:
Eq. 10
Using the results from the high pressure experiment (which are almost exactly linear), it is possible to calculate a value of hfg since it is analogous to the gradient of the graph in figure 7.
Gradient =
= Eq. 11
=
= 7657.568431 K
= 7657.568432 * 287 [7]
= 2198 kJ / kg (4sf)
According to published tables from theory and experiments performed more accurately than this one, the value of hfg kJ / kg for a temperature of around 373 K [5].
Since the values are in such close proximity, it can be seen that the results from this experiment appear to be reliable.
In the introduction, the aims of the experiment were to firstly that there is a relationship between pressure and temperature for this particular substance, and secondly that the relationship is accurately described by the Clausius-Clapeyron Equation.
Since the graphs show clear trends, in one case almost exactly linear, there is clearly some relationship between pressure and temperature. Further, that the value of hfg estimated from the results of this experiment is in close proximity to the value published in tables (which are known to follow the Clausius-Clapeyron trend) [6] the relationship in this experiment must also be described by the Clausius-Clapeyron equation.
Conclusions
- There is a definite relationship between pressure and temperature.
For the low pressure experiment, this is a curved relationship which is approximately linear.
For the high pressure experiment this relationship is almost exactly linear.
- Results would have been affected by experimental error, including errors in the equipment itself. The most important factor appears to be the escape of the substance from the equipment during measuring
- Human error would also be a factor in the results, where difficulty in making readings or parallax errors could affect their accuracy
- In the derivation of the equations used in the experiment, assumptions were made which were later shown to be valid
- The error in the experiment is estimated at 4.6%
-
Since the value of hfg estimated from the results is in close proximity to that published in tables, pressure and temperature indeed follow a trend described by the Clausius-Clapeyron equation
-
As such the objectives were met.
References
[1] http://www.chem.hope.edu/~polik/Chem345-1997/heatofvaporization/heatofvaporization.html
[2] http://www.science.uwaterloo.ca/~cchieh/cact/c123/clausius.html
[3] http://antoine.frostburg.edu/chem/senese/101/liquids/faq/clausius-clapeyron-vapor-pressure.shtml
[4] http://www.nyu.edu/classes/tuckerman/honors.chem/lectures/lecture_15/node2.html
[5] Thermodynamic and Transport Properties of Fluids
Rogers and Mayhew
Fifth Edition, 1995
[6] Thermodynamics Lecture Notes
J.W. Rose, 2004
[7] Mechanics of Fluids Lecture Notes
P.R. Wormleaton, 2005
