Resistance and conductivity

Two ways to carry a current

Join a copper wire to a battery and a lamp. The lamp glows. Now dip the two ends of the circuit into a glass of distilled water. The lamp stays dark. Stir in a spoon of salt. The lamp glows again.

What changed? The salt broke into ions. Sodium ions and chloride ions now move through the water, and they carry the current. A substance that gives ions in solution is an electrolyte.

• A strong electrolyte is almost completely in ions. Examples are sodium chloride, hydrochloric acid and potassium hydroxide.
• A weak electrolyte gives only a few ions. Acetic acid and ammonia solution are examples.
• A non-electrolyte, such as sugar, gives no ions. Its solution does not conduct.

A metal carries current in a different way, and the two ways behave differently:

• Metallic conduction. Electrons carry the current. The metal does not change. When the metal is heated, its atoms vibrate more and get in the way of the electrons, so it conducts less.
• Electrolytic conduction. Ions carry the current. A direct current changes the solution at the electrodes, because ions are discharged there. When the solution is heated, the ions move faster and the water flows more easily, so it conducts more. A weak electrolyte also splits into more ions when it is warmer.

Ions are much heavier than electrons and are slowed by the water around them. So even a strong electrolyte conducts far worse than a metal.

Resistance, conductance and conductivity

You know the resistance of a wire: $R = \rho\,\dfrac{l}{A}$. The length is $l$, the cross-section is $A$ and $\rho$ is the resistivity of the material. The same idea works for a column of solution between two plates. Then $l$ is the distance between the plates and $A$ is the area of the plates.

Three more words are useful for electrolytes:

• Conductance $G = \dfrac{1}{R}$. Its unit is the siemens, $1\ \text{S} = 1\ \Omega^{-1}$.
• Conductivity $\kappa$ (the Greek letter kappa) is $\dfrac{1}{\rho}$. Its unit is $\text{S/m}$.
• A large conductivity means a good conductor.

Put them together. We want $\kappa$, so we write

The conductivity is the conductance of a column of the solution that is $1\ \text{m}$ long and has a cross-section of $1\ \text{m}^2$. Chemists often use $\text{S/cm}$. Since $1\ \text{m} = 100\ \text{cm}$, we have $1\ \text{S/cm} = 100\ \text{S/m}$.

The cell constant

The ratio $\dfrac{l}{A}$ belongs to the cell, not to the solution. We call it the cell constant:

Its unit is $\text{m}^{-1}$, or $\text{cm}^{-1}$. Then the conductivity is

Here is a first example. A cell has two plates of area $2.0\ \text{cm}^2$, set $3.0\ \text{cm}$ apart. The cell constant is $G^* = \dfrac{3.0\ \text{cm}}{2.0\ \text{cm}^2} = 1.5\ \text{cm}^{-1}$, which is $150\ \text{m}^{-1}$. The cell is filled with a solution and reads $400\ \Omega$. The conductivity is $\kappa = \dfrac{150\ \text{m}^{-1}}{400\ \Omega} = 0.375\ \text{S/m}$.

Finding the cell constant

In a real cell, the plates are not perfect and the current spreads out at the edges. We cannot trust a measured $l$ and $A$. So we calibrate the cell. We fill it with a solution whose conductivity is known very well, and read the resistance. Potassium chloride is used because it can be made very pure. A solution of $0.100\ \text{mol/L}$ potassium chloride has $\kappa = 1.29\ \text{S/m}$ at $298\ \text{K}$.

We want $G^*$, so we write $G^* = \kappa R$. Say the cell reads $87.0\ \Omega$ with this solution. Then $G^* = 1.29\ \text{S/m} \times 87.0\ \Omega = 112\ \text{m}^{-1}$.

Now rinse the cell and fill it with the solution to be tested. It reads $241\ \Omega$. Then $\kappa = \dfrac{112\ \text{m}^{-1}}{241\ \Omega} = 0.466\ \text{S/m}$.

Why alternating current

Look at the figure. The supply is an alternating current of about $1000\ \text{Hz}$. This is not a detail. With a direct current the ions would be discharged at the plates. The solution near the plates would change, and the resistance would drift while you measure it.

An alternating current reverses about a thousand times each second. The ions only move to and fro by a tiny distance, and nothing is discharged. The two plates are covered with a rough black powder of platinum, called platinum black. It gives a very large surface, which keeps the effects at the plates small. The cell is one arm of a bridge circuit, and the resistance is found when the bridge balances. The cell also sits in a water bath, because the conductivity changes with temperature.

The sim shows $R = \dfrac{G^*}{\kappa}$. Double the conductivity and the resistance halves. A cell with a larger constant has a larger resistance for the same solution.

What decides the conductivity

The conductivity of a solution depends on four things:

• The number of ions in each cubic metre. More ions carry more charge. A strong electrolyte that is diluted has fewer ions in each cubic metre, so its conductivity falls.
• The charge on each ion. An ion with a double charge carries twice as much charge.
• How fast the ions move. This depends on the size of the ion, the solvent and the temperature. Hot solutions conduct better, by about 2 per cent for each kelvin.
• How much of the electrolyte is in ions. For a weak electrolyte this is only a small fraction.

Conductivity alone cannot tell strong and weak electrolytes apart, because it also depends on how much solute you took. The next lesson fixes this.

Practise with new numbers

Try it in Python

The program calibrates a cell with potassium chloride, then uses the cell constant to find the conductivity of a sample.

Common mistakes

• Mixing centimetres and metres. A cell constant in $\text{cm}^{-1}$ gives a conductivity in $\text{S/cm}$. Convert once, at the start.
• Dividing when you should multiply. To find the cell constant from calibration, multiply: $G^* = \kappa R$.
• Using a direct current. The plates would change the solution and the reading would drift.
• Calling the conductance the conductivity. The conductance belongs to this cell. The conductivity belongs to the solution and does not depend on the cell.
• Thinking an electrolyte conducts less when it is heated. That is true of a metal, not of a solution.

Remember these

• Strong electrolytes are almost all in ions. Weak electrolytes give only a few ions.
• Conductance $G = \dfrac{1}{R}$, in S. Conductivity $\kappa = \dfrac{1}{\rho}$, in S/m. $1\ \text{S/cm} = 100\ \text{S/m}$.
• Cell constant $G^* = \dfrac{l}{A}$ and $\kappa = \dfrac{G^*}{R}$.
• The cell constant is found with a solution of known conductivity, usually potassium chloride.
• An alternating current is used so that nothing is discharged at the plates.
• A metal conducts less when heated. A solution conducts more.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The same cell reads $250\ \Omega$ with solution P and $500\ \Omega$ with solution Q. Which solution has the larger conductivity, and by what factor?

2. Why is an alternating current used, and not a direct current, to measure the resistance of a solution?

3. A cell has a cell constant of $60\ \text{m}^{-1}$ and reads $400\ \Omega$. Find the conductivity in S/m and in S/cm.

4. A copper wire and a dish of copper sulphate solution, each joined to a battery, are both warmed. What happens to the current in each?