Sample Volume Equation Derivation
10, 100, and 350 cm3 Units
The AccuPyc II is a gas displacement pycnometer which measures the volume of solid objects of irregular or regular shape whether powdered or in one piece. A greatly simplified diagram of the instrument is shown below.
Sample volume is determined from the known cell and expansion volumes and from measured pressures.
The cell and expansion chamber are initially at ambient pressure and the valve is closed. The cell is then charged to an elevated pressure.
The ideal gas law gives:
|
(1) |
where
nc | = | moles of gas in the sample cell |
R |
= |
the gas constant |
T |
= |
temperature |
Vc |
= |
cell volume |
Vs |
= |
sample volume |
Vx |
= |
expansion volume |
P1 |
= |
initial cell pressure |
Expansion volume:
|
(2) |
where
Pa | = | ambient pressure |
nx |
= |
moles of gas in the expansion volume |
When the valve is opened, the pressure falls to an intermediate value, P2, and mass balance yields:
|
(3) |
Substituting from equations (1) and (2) into (3):
|
(4) |
or
|
(5) |
then
|
(6) |
Adding and subtracting Pa in the denominator and rearranging gives
|
(7) |
Dividing by (Pa - P2) in both the numerator and denominator
|
(8) |
or
|
(9) |
Using gauge pressure defined as:
|
(10) |
equation (9) is rewritten as:
|
(11) |
1 cm3 and 2000 cm3 Units
The 1 cm3 and 2000 cm3 pycnometers operate by filling the expansion chamber while the sample cell remains at ambient pressure. After P1 is equilibrated, the expansion valve opens to allow gas to expand into the sample chamber and P2 is equilibrated.
For the 1 cm3 and 2000 cm3 pycnometers, equation (1) becomes
and equation (2) becomes
A derivation similar to that above yields the working equation for the 1 cm3 and 2000 cm3 pycnometers
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