APPENDIX A
A simple model to describe fractionation processes in the ionisation chamber
In order to compare the experimental results with theory, we constructed a simple box
model that is described in this section. The ionisation chamber is represented by a
rectangular box with the following dimensions: 2 cm × 1.5 cm × 1 cm (volume V = 3.75
cm3, surface U = 15.5 cm2).
Since two different capillaries are connected to the mass spectrometer, one from
the sample open split and one from the reference open split, the flow rates and the shapes
of the CO2 peak in helium entering the ionisation chamber may be different. The shape of
a CO2 peak injected via the sample open split also depends on whether or not liquid
nitrogen is used to cyrofocus the CO2 and whether or not a gas chromatographic column
is used. To describe rectangular peak shapes a Fermi function (see Eqn. (1)) was chosen:
u
InFermi (t )   t t  w / v
m
e
1
(1)
where tm marks the centre of the peak, u controls the amplitude and the area, w describes
the width and the area of the peak, and v is the parameter determining the slopes. The
output flux Out(t) is defined by the amount of molecules N(t) and the mean residence
time  in the ionisation chamber. The mean residence time depends on the input flux
In(t) and the volume V of the ionisation chamber and can be calculated according to
Eqn. (2):

p
V
 MS
In(t ) p0
(2)
where pMS is the pressure in the ionisation chamber and p0 the standard pressure (1000
mbar). The pressure in the ionisation chamber is estimated to be about 1.2 mbar and, as a
result,  is calculated to be 0.98 s.
The adsorption flux Ad´(t) towards the walls of the chamber, given in molecules
per second, can be calculated according to the Hertz-Knudsen formula:
Ad '(t )  N (t ) 
U
kT

V 2 m
(3)
where U is the surface of the ionisation chamber, k is the Boltzmann constant, T the
temperature in Kelvin and m is the mass of the molecules in kg. Not every molecule that
hits the wall will stick. Therefore, the sticking probability S' needs to be implemented:
S '  (1  )2
(4)
which is, in a first approximation, dependent on the surface coverage (see Eqn. (5)) of
the walls. It is assumed that a CO2 molecule dissociates upon adsorption. Therefore, the
probability of two vacant sites occurring adjacent to one another is proportional to the
square of the concentration of vacant sites.
(t ) 
N ad (t )
N sat
(5)
where Nad(t) is the number of molecules adsorbed on the wall at a time t, and Nsat
corresponds to the number of molecules for a monomolecular layer. The latter can be
calculated geometrically by describing a close packing of spheres, resulting in Eqn. (6):
N sat 
U
2 3  rm2
(6)
where rm is the radius of the molecule (for CO2, rm is equal to 2.3110–10 m). For our
assumed geometry of the ionisation housing, the monomolecular layer consists of Nsat =
8.39·1015 CO2 molecules. As soon as a monomolecular layer is built up, the sticking
probability in our model is forced to zero. Therefore, the final function of the sticking
probability can be written as shown in Eqn. (7):
(1  ) 2 , N ad  N sat
S (t )  
 0 , N ad  N sat
(7)
The time-dependent flux Ad(t) towards the wall is then given by Eqn. (8):
Ad (t )  Ad '(t )  S (t )  N (t ) 
U
kT

 S (t ) (8)
V 2 m
The flux of ionised molecules is given by the ionisation efficiency and the
extraction efficiency of the ions through the acceleration lenses. As mentioned in the text,
the overall efficiency E for our mass spectrometer is 0.44‰.
The differential equations for the flow rates involved can then be written
according to Eqns. (9)–(12):
dN (t )
 In(t )  Ad (t )  Out (t )  Ion(t )
dt
(9)
dN ad (t )
U
kT
 Ad (t )  N (t )  
 S (t )
dt
V 2 m
(10)
dNOut (t )
N (t )
 Out (t ) 
dt

(11)
dN ion (t )
 Ion(t )  N (t )  E
dt
(12)
Since the walls are not free from CO2 when a new CO2 peak arrives, we insert a
number p that describes the initial condition for equation (10) according to Eqn. (13):
N ad (0)  p  N sat
(13)
where p is a number between 0 and 1. By choosing p = 0, the wall is free from adsorbed
molecules and, for p = 1, the wall is saturated. In order to fit the data best we had to
choose p between 0.996 and 1. Hence, the walls are always close to CO2 saturation. It
might be that part of this saturation of adsorbed molecules is non-CO2, e.g. CO or N2O,
etc. This number should be taken as an approximate value as it is difficult to have
accurate knowledge of the surface involved in the adsorption process. If the surface were
much smaller than assumed in the model, p would change accordingly. The issue of
greater interest is the amount of CO2 molecules that are adsorbed on the walls compared
with the number of CO2 molecules entering the ionisation chamber. Figure 12 describes
the area loss in percentage for an assigned peak area and for different values of the
parameter p as well as the fractionations of the 13C associated with a partial CO2
adsorption.
The question can be raised whether another flow rate ought to be added
describing the desorption process from the walls back to the gas phase. The amount of
CO2 which is adsorbed on the walls is assumed to be similar to the amount of CO2
desorbed from the walls between two consecutive CO2 peaks. By assuming that the
coverage of the walls was the same before and after a sample peak, one can roughly
calculate the desorption rate of CO2 to be (6.8 ± 1.4)·1010 molecules per second. The
molecules that are adsorbed on the wall vibrate with a frequency of the order ofv0 = 1013
s–1. For the desorption process a kinetic energy is required that is higher than the
desorption energy Edes. According to the Boltzmann distribution only the fraction
Nad·exp(-Edes/R·T) from Nad adsorbed molecules meet this requirement. Therefore, the
desorption rate des is given by Eqn. (14),
des  v0  N ad  e
 Edes
RT
 1013  N ad  e
 Edes
RT
(14)
where R is the universal gas constant, T the temperature in Kelvin and Edes the desorption
energy in J/mol (1 eV = 96.2 kJ/mol). With Eqn. (14) and Nad = 8.385·1015, calculated
with the model, one can calculate the desorption energy to be 1.7 eV for an assumed
temperature of 200°C. To estimate the influence of desorption, the mean residence time
of a molecule adsorbed on the wall can now be calculated. This time ad is given by
Eqn. (15):
 ad  1013  e

Edes RT

(15)
For the assumed temperature, Eqn. (15) results in a value of 1.2·105 s for ad. This
is compatible with our experience that it takes days to reach a low CO2 background after
CO2 admission. Therefore, as long as CO2 arrives in the ionisation chamber within
minutes to hours, we can neglect the desorption rate.
Since we are interested in the isotopic composition of the gas, corresponding
differential equations for the isotopic ratios have to be formulated (Eqns. (17)–(20)). Four
different fractionation factors i are used:
B/ A 
RB
RA
(16)
where RA is the isotopic ratio in a reservoir A and RB is the isotopic ratio in a reservoir B.
B/A describes the fractionation of molecules in the reservoir B with respect to that of
reservoir A. Fractionation can occur in the open split (, during the sorption process
(), during the pump down () of the gas in the ionisation chamber and during the
ionisation process ():

dR(t )
1 
 In(t )  1  Rin (t )  R(t )   Ad (t )  R(t )   2  1 




dt
N (t ) 
Out (t )  R(t )   3  1  Ion(t )  R(t )   4  1 


dRad (t )
1

 Ad (t )   2  R(t )  Rad (t ) (18)
dt
N ad (t )
(17)
dRout (t )
1

 Out (t )   3  R(t )  Rout (t ) (19)
dt
N out (t )
dRion (t )
1

 Ion(t )   4  R(t )  Rion (t ) (20)
dt
N ion (t )
This set of eight differential equations ((9)–(12) and (17)–(20)) can be solved
numerically using mathematical software. Within this study, 3 and 4 were always set to
1 assuming no fractionation during the pump down of the gas in the ionisation chamber
or during the ionisation process. It is possible that mainly during the pump down of the
gas fractionation occurs. Taking an upper limit for this fractionation factor of 1.00093 for
13
C, which corresponds to a fractionation due to diffusion of CO2 in helium, our model
suggests that this effect is smaller than 0.05‰ for areas larger than 2 Vs. This is within
the uncertainty of our measurements and therefore 3 is assumed to be 1. The model
output Rion(t), is finally expressed in the -notation relative to a reference gas with the
isotopic composition Rref:
 Rion (t )  3
 1 10
 R

 ref

 ion (t )  
(21)
APPENDIX B
Calculating flow rates with the Hagen-Poiseuille equation
The flow rate F in L/s through a capillary at standard temperature and pressure (STP)
can be calculated according to the Hagen-Poiseuille (Eqn. (22)):
F  1011 
 p  p2  [μl/sec] (22)
  r4
  p1  p2   1
8   l
2  p0
where p1 is the pressure in mbar at the beginning of the capillary, p2 is the pressure in
mbar at the end of the capillary, p0 is 103 mbar, r the radius in m, l the length of the
capillary in m, and  the viscosity of the gas in Ns/m2.
Estimation of the overall detection efficiency
The amount of CO2 molecules, NOpenSplit, entering the sample open split every second can
be calculated using Eqn. (23):
NOpenSplit 
F  N A  RCO2 / He
(23)
1012 V0
where NA is the Avogadro constant, V0 the molar volume of a gas in STP (22.41 l) and
RCO2/He the CO2-to-helium ratio in ppmv. Additional tests showed that the CO2/He ratio is
unchanged in the sample open split. Therefore, the molecule flow rate into the ionisation
chamber can be calculated according to Eqn. (24):
N IonChamber  NOpenSplit  f , f 
s
FMS
Fins
(24)
where f is the ratio of the flow rate from the sample open split to the mass spectrometer to
the input flow rate through the sample capillary to the sample open split. The number of
CO2 molecules detected per second on the Faraday cups can be calculated with Eqn. (25):
NCup 
A
10  R  q
3
(25)
where R is the resistance of the Faraday cup at m/z 44 (R = 3·108 q is the elementary
charge, and A is the signal amplitude at mass-to-charge ratio 44 in mV measured on the
Faraday cup. The overall efficiency E of the ionisation and the extraction process can be
investigated by comparing the numbers of CO2 molecules entering the ionisation
chamber with the numbers of CO2 molecules detected on the Faraday cup.
E
NCup
(26)
N IonChamber
We found E to be 0.44‰.
Fractionation by diffusion
The fractionation factor of the CO2 diffusion in helium can be calculated according to
Eqn. (27):
M
M

CO2
*
CO2
 M He
 M
 M
*
 M He
CO2
(27)
CO2
where *M CO2 is the molar mass of the heavier CO2 isotope (45 g/mol or 46 g/mol),
M CO2 is the molar mass of CO2 (44 g/mol) and M He is the molar mass of helium (4
g/mol). The same formula can be used to calculate the fractionation factor for the
diffusion of CO2 in CO2. This leads to the fractionation factors given in Eqn. (28):
 CO  He  1.00093,
13
 CO  He  1.00181,
18
13
2
18
2
 CO CO  1.00560
2
2
 CO CO  1.01105
2
2
(28)