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Influence of diffusion on the measurement of low pressure with the McLeod vacuum gauge. Based on a paper by Gaede
VACUUMVoI. 13, pp. 579-581. Pergamon Press Ltd.
Printed in Great Britain.
Influence of Diffusion on t h e M e a s u r e m e n t McLeod V a c u u m
Gauge.
of Low Pressure w i t h t h e
Based on a P a p e r b y Gaede
Ca. M E I N K E AND G. R E I C H
E. Leybolds Nachfolger, K61n-Bayenthal, Germany In a fundamental paper on the diffusion o f gases in mercury vapour, Gaede concludes, amongst other things, that pressure measurements carried out with the McLeod gauge are too low i f a cold trap has been inserted between the gauge and the recipient. In the following, Gaede's line o f thought, as far as it concerns the above phenomenon, is reproduced and some experimental results o f very recent date confirming his theory are given.
IN the year 1915, Gaede published in the Annalen der Physik a paper on " T h e diffusion of gases through mercury vapour at low pressures and the diffusion pump ,,1. In this publication Gaede criticised the opinion generally held at that time that a diffusion process necessitates in all cases that the sum of the partial pressures is constant in every part of the system. In other words, for a 2-component system we must have Pl + P1 = P 2 + P2.
Inserting the u of the above equation in eqn. (3) we obtain log e 7 = lQ/Doq.
(1)
log e Y = lL(Pd + Pl -- P2)/Doq. (5) In the modern notation for L, which differs slightly from that of Gaede, we have
F r o m this it was concluded that the total pressure in a receiver under evacuation by a diffusion pump could not drop below the saturation vapour pressure of the pumping medium2. In a simple experiment Gaede caused mercury vapour to flow through a tube into a condenser and investigated the diffusion of air into the vapour stream. He measured the air pressures Pl and P2 at the- beginning and end of the tube respectively and determined the pressure Pd of the mercury vapour from the saturation vapour pressure at the temperature of the mercury. The vapour pressure of the mercury in the condenser is practically zero. He found that the total pressures at the beginning and end of the tube are different, i.e., under the condition of diffusion in a steady stream ofvapour, eqn. (1) no longer applies. In order to express the experimental results in a quantitative manner, Gaede started with the diffusion equation of Stephan which holds for the dynamic diffusion equilibrium in the vapour stream :
dp dx where
p
: (ppau)/Do,
r3 l~rr (P_d + Pl -- P2"~
L=~ t~kfnr['pd+Pl--P2)
8 /~RT
l
- l + i~¢ ZMj • 8
/Ri
t{P_d+Pl--P2~(6)
Gaede applied this expression to various types of diffusion pumps. In passing he also mentions the importance of this effect on the recording of low pressures in a receiver with a McLeod gauge, when a cold trap is situated between the gauge and the receiver. Such an arrangement constitutes to all intents a diffusion circuit in which the mercury vapour from the mercury storage vessel flows through the gas to be measured into the cold trap, taking some gas from the measuring bulb of the gauge with it. A numerical example is given : The receiver contains air at low pressure which is to be measured with a McLeod gauge. If we neglect the first summation term in (6) which corresponds to the share of Poiseuille Flow in the process and can therefore be omitted at low pressures, we obtain
(2)
-- pressure of gas, = velocity ofvapour,
(8 /RT 1
Do : diffusion coefficient at a pressure of 1 torr.
=exp \3~/2~-Af/D0Pd r
integrating over the length of the tube log e P2/Pl : log e T = ( P d 1 u)/Do
2-
with q = nr2, eqn. (5) now becomes
Pd -- pressure offlowingvapour, u
(4)
Since Q = L(Pd + Pl -- P2) wbero L = conductance of tube, Pd + Pl = total pressure at beginning of tube, P2 = total pressure at end of tube. Equation (4) becomes
)
,
(6a)
where (P2 - pl) is also assumed negligible at low pressure in eqn. (6).
(3)
The quantity of mercury vapour Q flowing through the crosssection q of the tube per unit time is given by
Now at 18°C,
Do ~ 91.2 cmZ/sec, torr Pd ~ 10-3 torr.
Q = uqpd 579
580
CH. MEINKE and G. REXCH
Thus we obtain finally
? = exp (. 13r)
were always higher than those determined by an expansion method. A n explanation of this deviation was possible with the help of the Gaede formulation.
: 1 q- .13r q- .. where r is in cm. From this follows : " If for example a McLeod gauge is joined to an absorption vessel immersed in liquid air, the radius of the connecting pipe r being 0.4 cm, then the partial pressure P2 of the air in the receiver connected to the cooling vessel is ,~ 5 per cent. larger than the partial pressure Pl of the air in the McLeod gauge".
A sketch of the experimental apparatus is shown in Fig. 1. From what has previously been said, it follows dearly that a lower pressure exists at the McLeod gauge than behind the cold traps where the ionization gauge is situated. 1.6
In sentences following on the above, Gaede pointed out that the effect is very much less if instead of air, the receiver contains hydrogen. These facts, already fully appreciated by Gaede at that time, were subsequently overlooked. It is only quite recently when interest in more accurate pressure measurements has arisen and the possibility of greater accuracy appeared possible, that discrepancies in the pressure measurement by this method were established. Thus for example Podgurski and Davies3 noted during an investigation on the accuracy of the McLeod gauge a pressure rise of 6 :E 2 per cent for xenon and 2 :E 1 per cent for argon on the side of the trap free of mercury vapour. They refer in this connection to Dushman4 who reproduces Gaede's lines of reasoning in part but does not point out its significance for pressure measurements. Isshii and NakayemaS, when calibrating ionization manometers, noted a strong dependence of the sensitivity on the ambient temperature. They explain the change in sensitivity with the help of Gaede's equations and established a relatively good agreement of their experimental results with those calculated by Gaede's method. The pressure effect to be expected from eqn. (5) and (6) was also confirmed qualitatively. They further deduced the dependence on r.
----x--
Experimenl Theory
1.4
>..
/
~1~- 1 . 3 -
--
[
1.2
¢
/
/~////
T=exp (Pd L q-~o )
r/x
j
,~.
I0
20
30
Temperature,
°C
FIG. 2. Pressure measurement with McLeod gauge and cold trap. ~, = f(r). A quantitative investigation of the effect was undertaken, starting with eqn. (5). Figure 2 shOws the results of an investigation on the effect of temperature, the full line indicating the theoretical, and the dotted line the experimental results, Figure 3 shows corresponding curves for the pressure effect for various gases. Equation (6) shows that ~,, whilst being a function of the radius of the connecting tube, is independent of its length.
The authors of the present paper reported6 on a comparison of methods of calibration for the ionization manometer. In the course of these calibrations, they noted that the sensitivities determined with the help of the McLeod gauge
Xenon 1.50 ~
~
-
1.40
I
[ [ I
T=25"C
®, I
I
I I 1
1'30 •
Q •
i,
Argon
0 '0
1.20
J
-
Bokoble
I'10 1"08 1'06 Helium 1.0,1 t-02
x
1.°°lb-~
io-~
×
I
x
\~o. He
Jo-~ Pressure,
FIO. 1.
x
x
jo-~
Io
A
ton"
Fla. 3. Pressure measurement with McLeod gauge and cold trap, y as a function of nature of gas and pressure.
Influence of diffusion on the measurement of low pressure with the McLeod vacuum gauge. These relationships were e x a m i n e d b y us qualitatively. F o r a m o r e detailed description o f e x p e r i m e n t a l c o n d i t i o n s a n d discussion o f results we refer to o u r paper7. T h e curves o f this r e p o r t s h o w t h a t G a e d e ' s line o f t h o u g h t regarding t h e possibility o f errors in the pressure r e c o r d e d o f a M c L e o d gauge was substantially correct.
Based on a paper by Gaede
581
References 1 W. Gaede, Ann. Phys., 46, (1915), 354-392. 2 Lehrbuch der Physik (Textbook of Physics), Mfiller-Ponillet, (1906), p. 505. 3 Podgurski and Davies, Vacuum, 10, (1960), 377. 4 Dushman. Vacuum Technique, John Wiley, (1955), p. 176. 5 Isshii and Nakayema, 1961 Trans. 8th Vac. Syrup. & 2nd lnternat. Congr., Pergamon Press, (I 962). 6 Meinke and Reich, Vakuumtechnik, 11, (19~2~, 86. 7 Meinke and Reich, Vakuumtechnik, 12, (3), (1963), 79.
Printed in Great Britain.
Influence of Diffusion on t h e M e a s u r e m e n t McLeod V a c u u m
Gauge.
of Low Pressure w i t h t h e
Based on a P a p e r b y Gaede
Ca. M E I N K E AND G. R E I C H
E. Leybolds Nachfolger, K61n-Bayenthal, Germany In a fundamental paper on the diffusion o f gases in mercury vapour, Gaede concludes, amongst other things, that pressure measurements carried out with the McLeod gauge are too low i f a cold trap has been inserted between the gauge and the recipient. In the following, Gaede's line o f thought, as far as it concerns the above phenomenon, is reproduced and some experimental results o f very recent date confirming his theory are given.
IN the year 1915, Gaede published in the Annalen der Physik a paper on " T h e diffusion of gases through mercury vapour at low pressures and the diffusion pump ,,1. In this publication Gaede criticised the opinion generally held at that time that a diffusion process necessitates in all cases that the sum of the partial pressures is constant in every part of the system. In other words, for a 2-component system we must have Pl + P1 = P 2 + P2.
Inserting the u of the above equation in eqn. (3) we obtain log e 7 = lQ/Doq.
(1)
log e Y = lL(Pd + Pl -- P2)/Doq. (5) In the modern notation for L, which differs slightly from that of Gaede, we have
F r o m this it was concluded that the total pressure in a receiver under evacuation by a diffusion pump could not drop below the saturation vapour pressure of the pumping medium2. In a simple experiment Gaede caused mercury vapour to flow through a tube into a condenser and investigated the diffusion of air into the vapour stream. He measured the air pressures Pl and P2 at the- beginning and end of the tube respectively and determined the pressure Pd of the mercury vapour from the saturation vapour pressure at the temperature of the mercury. The vapour pressure of the mercury in the condenser is practically zero. He found that the total pressures at the beginning and end of the tube are different, i.e., under the condition of diffusion in a steady stream ofvapour, eqn. (1) no longer applies. In order to express the experimental results in a quantitative manner, Gaede started with the diffusion equation of Stephan which holds for the dynamic diffusion equilibrium in the vapour stream :
dp dx where
p
: (ppau)/Do,
r3 l~rr (P_d + Pl -- P2"~
L=~ t~kfnr['pd+Pl--P2)
8 /~RT
l
- l + i~¢ ZMj • 8
/Ri
t{P_d+Pl--P2~(6)
Gaede applied this expression to various types of diffusion pumps. In passing he also mentions the importance of this effect on the recording of low pressures in a receiver with a McLeod gauge, when a cold trap is situated between the gauge and the receiver. Such an arrangement constitutes to all intents a diffusion circuit in which the mercury vapour from the mercury storage vessel flows through the gas to be measured into the cold trap, taking some gas from the measuring bulb of the gauge with it. A numerical example is given : The receiver contains air at low pressure which is to be measured with a McLeod gauge. If we neglect the first summation term in (6) which corresponds to the share of Poiseuille Flow in the process and can therefore be omitted at low pressures, we obtain
(2)
-- pressure of gas, = velocity ofvapour,
(8 /RT 1
Do : diffusion coefficient at a pressure of 1 torr.
=exp \3~/2~-Af/D0Pd r
integrating over the length of the tube log e P2/Pl : log e T = ( P d 1 u)/Do
2-
with q = nr2, eqn. (5) now becomes
Pd -- pressure offlowingvapour, u
(4)
Since Q = L(Pd + Pl -- P2) wbero L = conductance of tube, Pd + Pl = total pressure at beginning of tube, P2 = total pressure at end of tube. Equation (4) becomes
)
,
(6a)
where (P2 - pl) is also assumed negligible at low pressure in eqn. (6).
(3)
The quantity of mercury vapour Q flowing through the crosssection q of the tube per unit time is given by
Now at 18°C,
Do ~ 91.2 cmZ/sec, torr Pd ~ 10-3 torr.
Q = uqpd 579
580
CH. MEINKE and G. REXCH
Thus we obtain finally
? = exp (. 13r)
were always higher than those determined by an expansion method. A n explanation of this deviation was possible with the help of the Gaede formulation.
: 1 q- .13r q- .. where r is in cm. From this follows : " If for example a McLeod gauge is joined to an absorption vessel immersed in liquid air, the radius of the connecting pipe r being 0.4 cm, then the partial pressure P2 of the air in the receiver connected to the cooling vessel is ,~ 5 per cent. larger than the partial pressure Pl of the air in the McLeod gauge".
A sketch of the experimental apparatus is shown in Fig. 1. From what has previously been said, it follows dearly that a lower pressure exists at the McLeod gauge than behind the cold traps where the ionization gauge is situated. 1.6
In sentences following on the above, Gaede pointed out that the effect is very much less if instead of air, the receiver contains hydrogen. These facts, already fully appreciated by Gaede at that time, were subsequently overlooked. It is only quite recently when interest in more accurate pressure measurements has arisen and the possibility of greater accuracy appeared possible, that discrepancies in the pressure measurement by this method were established. Thus for example Podgurski and Davies3 noted during an investigation on the accuracy of the McLeod gauge a pressure rise of 6 :E 2 per cent for xenon and 2 :E 1 per cent for argon on the side of the trap free of mercury vapour. They refer in this connection to Dushman4 who reproduces Gaede's lines of reasoning in part but does not point out its significance for pressure measurements. Isshii and NakayemaS, when calibrating ionization manometers, noted a strong dependence of the sensitivity on the ambient temperature. They explain the change in sensitivity with the help of Gaede's equations and established a relatively good agreement of their experimental results with those calculated by Gaede's method. The pressure effect to be expected from eqn. (5) and (6) was also confirmed qualitatively. They further deduced the dependence on r.
----x--
Experimenl Theory
1.4
>..
/
~1~- 1 . 3 -
--
[
1.2
¢
/
/~////
T=exp (Pd L q-~o )
r/x
j
,~.
I0
20
30
Temperature,
°C
FIG. 2. Pressure measurement with McLeod gauge and cold trap. ~, = f(r). A quantitative investigation of the effect was undertaken, starting with eqn. (5). Figure 2 shOws the results of an investigation on the effect of temperature, the full line indicating the theoretical, and the dotted line the experimental results, Figure 3 shows corresponding curves for the pressure effect for various gases. Equation (6) shows that ~,, whilst being a function of the radius of the connecting tube, is independent of its length.
The authors of the present paper reported6 on a comparison of methods of calibration for the ionization manometer. In the course of these calibrations, they noted that the sensitivities determined with the help of the McLeod gauge
Xenon 1.50 ~
~
-
1.40
I
[ [ I
T=25"C
®, I
I
I I 1
1'30 •
Q •
i,
Argon
0 '0
1.20
J
-
Bokoble
I'10 1"08 1'06 Helium 1.0,1 t-02
x
1.°°lb-~
io-~
×
I
x
\~o. He
Jo-~ Pressure,
FIO. 1.
x
x
jo-~
Io
A
ton"
Fla. 3. Pressure measurement with McLeod gauge and cold trap, y as a function of nature of gas and pressure.
Influence of diffusion on the measurement of low pressure with the McLeod vacuum gauge. These relationships were e x a m i n e d b y us qualitatively. F o r a m o r e detailed description o f e x p e r i m e n t a l c o n d i t i o n s a n d discussion o f results we refer to o u r paper7. T h e curves o f this r e p o r t s h o w t h a t G a e d e ' s line o f t h o u g h t regarding t h e possibility o f errors in the pressure r e c o r d e d o f a M c L e o d gauge was substantially correct.
Based on a paper by Gaede
581
References 1 W. Gaede, Ann. Phys., 46, (1915), 354-392. 2 Lehrbuch der Physik (Textbook of Physics), Mfiller-Ponillet, (1906), p. 505. 3 Podgurski and Davies, Vacuum, 10, (1960), 377. 4 Dushman. Vacuum Technique, John Wiley, (1955), p. 176. 5 Isshii and Nakayema, 1961 Trans. 8th Vac. Syrup. & 2nd lnternat. Congr., Pergamon Press, (I 962). 6 Meinke and Reich, Vakuumtechnik, 11, (19~2~, 86. 7 Meinke and Reich, Vakuumtechnik, 12, (3), (1963), 79.