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Then assuming the heat capacity of the solute is negligible,
Hd0 = (Heat Evolved in Cell)- (Heat Convected from Cell)
- (Heat Conducted from Cell).
where, H is the heat capacity of the cell,
i.e H= KmdmSm + ^sdsSs + ÊgdgSg
and l/m, l/s, and l/g are the volume of mobile pnase in the cell, the
volume of absorbant in the cell and the wall volume respectively,
dm,ds, and dg are the densities of the mobile phase, absorbent and cell walls respectively,
Sm, Ss, and Sg are the specific heats of the mobile phase, absorbent and cell walls respectively.
Note the italic form, ( Ó), is used to distinguish it from volume, (V), and plate volume, (v).
The heat evolved in the cell during the passage of a volume (dv) of mobile phase will now be derived. Let a volume (dv) of mobile phase, equivalent to (cadv) ml, enter the cell, and let the concentration of solute (a) in the incremental volume be (Xn). Let an equivalent volume (c8dv) of mobile phase be displaced from the cell, and let the concentration of solute in the mobile phase contained by the cell prior to the introduction of the volume (dv) be (Xm)-
Now the net change of mass of solute (dm) in the cel) wilt be,
dm = (Xn-Xm)cadv ........................................................ (21)
As equilibrium is assumed to occur in the detector celi the introduction of the mass of solute (dm) will result in a change in concentration of solute in the mobile phase and adsorbent of (dXm), and (dXs) respectively, where (Xs) is the concentration of solute in the absorbant.
Thus, dm = l/s dXs + l/mdXm
and as, dXs = KdXm
dm = ÓsKdXpn + VmdXm
= ( (/st< + l/m)dXm ............................................... (22)
Equating equations (21) and (22) and rearranging,
>'sK+ jfi ildXm
Ca J dv
Now, l/sK + Óò is the 'effective cell volume' in much the same way that (ca) is the column 'plate volume'.
(KJ's+^m) "effective plate volume"
Let, --- = Ca =---for solute (a)
ca "column plate volume'
Thus, Ca— + Xm = Xn ........................................................ (23)
Multiplying throughout by (eCa),
reca a dv
+ Xme a
CaeC0Xm = |xneCadv +R
Now, when v = 0 the solute has not moved from the point of injection on the column.
Thus, when v = 0 ,Xm = Xn = 0 and consequently, R = 0 and,
e^a dv ....................................... (24)
Equation (24) provides an expression for (Xm). Continuing; if the change in mass of solute on the absorbant due to a volume flow of mobile phase cadv, is (dms), then the consequent heat evolved dG in the cell will be given by:-
where (g) is the heat of adsorption of the solute in cal. per gram of solute.
dG=g Vs dG = KgVs
dv and, asdXs = KdXm.
From equation (23),
Substituting for ( ^ ), dv
Thus, substituting for (Xm) from equation (24) and rearranging,
fKgvs] t Xn- \ V V —leCa fe c® Xndv
i ca J ê [CaJ J j
-v ~v / >
e_vvn 1 n!
-V / \
ñ» I e-vvn e 81--
dv = f(v)
dG= Kg^f(v)dv .........................................(25)
Now, the heat conducted from the cell will be considered to be controlled by the radial conductivity of the total cell contents and not by the cell walls alone. Furthermore, the axial conductivity of the cell will be ignored as its contribution to heat loss will be several orders of magnitude less than that lost by axial convection. Consequently, as the cell is cylindrical, the heat conducted radially from the cell has been shown to be (20)
where, (E) is the thermal conductivity of the cell contents,
(0) is the excess temperature of the cell above its surroundings,
(rt) is the radius of the sensing element and (dt) is the time taken for a volume cadv of mobile phase to pass through the cell.
Now (dt) refers to the time interval during the introduction of the volume (cadv) of mobile phase and thus, if the flow rate is (Q),
u t ^ ^ ^ ^ ¦ 2nlEca8dv Thus, heat conducted from the cell is, -9—
and the heat convected from the cell is, dmSm8cadv
Thus, inserting the above expressions for the heat conducted from the cell and the heat convected from the cell, together with the heat evolved from the cell, from equation (25) in equation (20A),
liultiplying equation (26) throughout by e H and integrating,
Now, on injection of the sample, when v=0, 0 = 0 and f(v) = 0 and consequently, R= 0,
v v v -pca- f pca- A
Therefore, 0 = e H0je H-f(v)dv ......................(27)
Letting — — ip and — = ô
a H Y H
Then, 0V = <pe_*v Je*vf(v)dv
It is seen from equation (28) that the constant (q>) merely effects the magnitude of (0) but the constant (ô) and f(v) condition the shape of the temperature profile and produces the curious shaped peaks recorded by the detector. The constant (ô) can be considered as the heat loss factor of the cell. It should be noted that the magnitude of f(v) will depend on the value of (Ca) the ratio of the ’effective volume of the cell’ to the 'plate volume' of the column.