ELECTROLYTIC CONDUCTIVITY

A) Theoretical
	The electric resistance R of a uniform conductor is directly proportional to its length l and inversely proportional to its cross-sectional area A:
									1)
where the proportionality constant  is called resistivity and its inverse  is called conductivity. The unit for the latter is 1/m=S/m=mho/m.
	Electric conductivities range from 108 S/m (for a metallic conductor at room temperature) to 10-15 S/m (for an insulator like SiO2).
	As an electrolyte solution has no free electron, its conductivity is made up of contributions from each ions present. For instance, we are given a binary electrolyte of composition AnBm which has fully dissociated to ions Ak+ and Bl-. Its conductivity is:

			 = F cA k uA  +  F cB l uB					2)

where uA and uB are the electric mobilities of ions A and B, respectively. (Electric mobility ui is the drift velocity vi of ion i divided by the driving electric field strength E : ui =vi/E.) 
	The mass balances for the dissociation process are:

			cA  =  n c,		cB  =  m c				3)

where c is the gross (or initial) concentration of AnBm (in mol/m3).
	By the well-known condition of electroneutrality

			k n  =  l m    						4)

where  is the electrochemical valence of AnBm . Dividing both sides of Equation 2) by c and taking into account Equations 3) and 4) one gets:

							5)

where ,  A  and  B are the molecular (or molar) conductivities of  AnBm and ions Ak+  and Bl-, respectively.

A note: molecular conductivity is frequently confused with equivalent conductivity designated by , too. Equivalent conductivity reads as:
				6)
and letter  designates the equivalent conductivities of  the referred components in this case. We will avoid this problem by using only electrolytes of =1.

	The order of magnitude of molecular conductivity is 10-3 Sm2/mol for ions in aqueous solutions, except for ions H3O+ and OH- whose are ten times more. These exceptionally large values are due to the hydrogen bonded structure of water as that makes possible for ions H3O+ and OH- to take part in conduction without their movement. They can join one end of a serial of hydrogen bonded water molecules, transfer their charge through the serial (by means of rearrangements of hydrogen bonds) to the water molecule being on the other end, finally the latter molecule produces another free H3O+ or OH-, respectively.
	The value of  is important for technology while  has its own role in physico-chemical treatments. 
	The dependence of  and  on concentration c can be studied on Figures 1 and 2.

			Figure 1					Figure 2

	One can see the serious differences between the strong (that is fully dissociated) and the weak (that is partially dissociated) electrolytes. It is worth to notice 
		i)	the linear dependence of  and the almost independence of  on 			concentration for dilute strong electrolytes;
		ii)	 tends to its maximal value () as c goes to zero for any 				kind of electrolytes.
	This is due to the fact that ions always decrease each other's field strength unless their distance is infinite. Clearly, that can happen only in an infinitely dilute solution where the activity coefficient  is exactly one by definition.
	An average electric charge density, consequently the divergence of an effective  field strength in an electrolyte solution can be characterized by square root of ionic strength I:

				 						7)

	As ionic strength increases  decreases, according to the Debye-Hückel theory, in the region I1 mol/m3 :

									8)

where multiplier C is constant for a given electrolyte-solvent pair. (Index  shows that there is only one activity coefficient in the solution, as there is only one average electric field strength, and it is valid for each cation and anion.)
	For weak electrolytes, the effect caused by decrease in degree of dissociation  highly surmounts that caused by decrease of  :

			 ,	or					9)

	An important consequence of Equations 7)-9) is that the   - c1/2   graph of strong electrolytes must be a straight line of small negative slope (almost horizontal) and that of weak ones must be of big negative slope, whether straight or curved.
	As the mechanism of any transport process in solution is jumping of particles from their position into a neighbouring vacancy, and the number of vacancies (the mobility of ions is proportional to it) is governed by the Maxwell-Boltzmann distribution, the molecular conductivity has a strong temperature dependence. For strong electrolytes:

			 							10)

where E is the energy needed to create a vacancy (otherwise: activation energy of conductance).
The dependence of  on temperature for weak electrolytes is even stronger because the dissociation constant is temperature dependent, too!

B) Equipment
	Conductance of a material body can be measured by taking a piece of known length and cross sectional area from it and measuring the current that a given voltage drives through it. Clearly, we have to immerse at least two electrodes into the electrolyte. Alas, there are some phenomena on the electrode-electrolyte interfaces which produce voltages, too. Their value must be decreased to be negligible in a good equipment:
i)	Equilibrium electrode potentials; their difference can be measured between electrodes in absence of current. The equipment must have identical electrodes to get rid of this phenomenon.
ii)	Polarization (so called overpotential) that is departure of electrode potential from the equilibrium value  due to the current through the interface. Overpotential, for not too large currents, depends linearly on current (polarisation resistance is ohmic) and inversely on surface area of the electrode-electrolyte interface. Consequently, the electrodes must have as large surfaces as possible.
iii)	Electrochemical double layer, due to the adsorbed ions on the surface of the electrode and the counter-ions gathered close to it. They form a capacitor whose impedance is connected in parallel to the polarisation resistance. If we use AC of high enough frequency then the capacitor of the double layer can short-circuit the polarisation resistance.
	Problems mentioned above under ii) and iii) are nonexistent if we apply, instead of two electrodes, four ones. The two outer electrodes are to conduct current while voltage drop can be measured, practically without current through them, between the two internal ones. Of course, the two outer electrodes are polarized but the extent of this phenomenon is not interesting in this case; even DC can be used.
	We may immerse a bell-shaped electrode (Figure 3) into the electrolyte solution. That has three platinum rings covered by electrodeposited platinum black. (The three electrodes form two poles as the two outer ones are connected.) The glass rode wearing the platinum rings is surrounded by a glass bell in order to possess a firm cell constant. Kcell

										11)

and to exclude the influence of convectional streams.

	Figure 3					Figure 4

	The electrode and a device called conductivity meter are connected by a shielded cable. The functional circuit diagram of the system can be seen on Figure 4. The alternating current i driven by voltage generator Ug of the conductivity meter is practically proportional to conductance G=1/R, and voltage meter V measuring the voltage drop on Rm can be calibrated to G if

			 						(12)
										(13)
										(14)

where X1 and X2 are the impedances of electrode-electrolyte interfaces Ch1-Rh1 and Ch2-Rh2 , respectively, XC is the impedance due to the capacitance CC of the shielded cable.
	As XC = 1/C ( is the angular frequency and C is the capacitance), and C is proportional to the surface area of the condenser plate, we are interested in increasing both the the surface area and the measuring frequency. This is the reason of using platinated platinum electrodes instead of flat ones. Of course, seek for a higher frequency is limited by condition in Eq. 14) to be fulfilled.
	Otherwise, we may immerse a conductivity cell having four electrodes. The firm cell constant is provided by the tubular structure (with two larger lateral apertures) of the cell. The four graphite electrodes are on the internal surface of the tube. In addition, this cell has temperature sensors, too. 
	The functional circuit diagram of the conductivity meter belonging to the four electrodes cell differs from the one on Figure 4 in one respect only: there is no Rm, so voltage drop is measured between the two internal electrodes with a very high input impedance voltage meter.

C) Experimental
	The aim of our work is to decide whether our electrolyte weak or strong is. 
	In order to do this we measure conductance at different concentrations of the electrolyte, calculate conductivities and molecular conductivities and plot them to see their tendency as concentration changes. 
	1. Change the distilled water in the test tube the electrode immersed in again and again while the measured conductance decreases.
	2. Pipette 20.00cm3 distilled water into the test tube, immerse the electrode and allow them to reach thermal equilibrium with the thermostating bath (approximately 15 minutes). Note the temperature and the conductance of distilled water.
	3. With a burette or a metering pipette, add the following amounts to the thermostated distilled water from the stock solution: 0.1cm3, 0.1cm3, 0.2cm3, 0.2cm3, 0.4cm3, 1.0cm3, 1.0cm3, 2.0cm3. Gently mix the solution with the electrode after each addition and read conductance. Subtract the conductance of distilled water from it.
	4. Clean the system again as written in Paragraph 1 and fill the test tube with 0.01 mol/dm3 KCl solution. (Its conductivity is =0.1408 S/m. on 25.0C.) Read conductance and subtract the conductance of distilled water from it. Calculate Kcell with the following equation:

				 = Kcell G						15)

With a four electrode cell, this procedure is needless. The cell constant is factory-tuned and stored in the digital conductivity meter, so immediately conductivity can be read on the display.

	5. Supposing additivity for volumes, calculate the concentration for each measurement. Calculate conductivities using Eq. 15) and molecular conductivities using Eq. 5).
	6. Plot the graphs   - c  and   - c1/2.  Judging from them, decide whether your electrolyte strong or weak is. If you find it to be strong, read  by linear extrapolation on graph  - c1/2. If you find it to be weak, calculate  from contributions of its ions using Eq. 5).
	7. Calculate dissociation constant Kad of the weak electrolyte. First calculate degree of dissociation  (Eq. 9)), value of Kcd (Eq. 16)) and ionic strength I (I = c  for =1 electrolytes) for each concentration.

								16)

As Kcd is connected to the thermodynamically significant Kad  by Equation 17):

					17)

and  is connected to I by Equation 8), plot the graph  .  (It must be a straight line of small positive slope.) Read the logarithm of Kad by extrapolation to I=0.