Cell Potential: The Nernst Equation - Online Tutor, Practice Problems & Exam Prep

Understanding cell potential is crucial in electrochemistry, particularly when examining how it correlates with the Nernst equation. The reaction quotient (Q) plays a pivotal role in this relationship, as it reflects the ratio of product to reactant concentrations at a given moment, excluding solids and liquids. This quotient is vital for determining the maximum potential of an electrochemical cell at the instant the circuit is established, as voltage tends to decrease over time due to cell degradation.

The standard cell potential (E°_cell) is calculated under standard conditions, where ion concentrations are 1 molar, pressure is at 1 atmosphere, and pH is 7. However, the Nernst equation becomes essential when conditions deviate from these standards. The Nernst equation, given by:

$E_{\mathrm{cell}}=E_{°}{\mathrm{cell}}_{}-\frac{0.05916}{n}\times log\left(Q\right)$

where n is the number of moles of electrons transferred in the redox reaction, allows for the calculation of non-standard cell potentials. This equation adjusts the cell potential for different ion concentrations, providing a more accurate measure of a cell's voltage under varying conditions.

The Reaction Quotient

Now, before we can talk about cell potential and its connection to the Nertz equation, it's important to revisit the idea of our reaction potion. Now recall that the reaction potion which uses the variable Q, is a ratio of product to react and concentrations at a particular time, and we're going to say it can be calculated by setting up an expression and ignoring solids and liquids.

Remember, like your equilibrium constant K, we only care about gaseous and aqueous aqueous species. Now how does this relate to electoral count? Well, we're going to say here that for electrochemical sounds it helps to find the Max potential at the exact moment the cell serpent is connected because as we know overtime our voltage is going to decrease as our battery, as our electrochemical cell overall degrades, right.

So this is where the reaction quotient comes into play. It helps us to find that exact moment in which our voltage is going to be at its highest.

The Reaction Quotient Example

Here it says what is the reaction quotient for the following redox reaction with the given concentrations. Alright, so here we have lead to ion reacting with two moles of potassium solid to produce one mole of lent solid +2 moles of potassium ion. Here we're going to say that the concentration of lead to ion and potassium ion are given as these molarities.

Now remember that your reaction quotient Q can be solved by setting up its expression and Q equals products overreacts. Now remember with this expression we ignore solids and liquids, so the solids we're going to ignore. So it basically becomes the concentration of K positive. And remember, because there's a 2 here, that becomes our exponent, so K positive squared divided by PB2 plus.

Here we plug in the values for each one. So potassium ion it's 0.0015, which will be squared divided by 0.0880. When we punch this into our calculators, we get 2.6×10-5. So this represents the value for our reaction quotient Q.

The Nernst Equation

Now recall that our standard cell potential is calculated when ions in half cells have values of one molar, one atmosphere and pH equal to 7. Now the Nernst equation is used to find the cell potential when ion concentration or concentrations do not equal 1 molar. So that's what we're going to use our Nernst equation because it's going to help us to calculate nonstandard cell potentials.

Now here the Nernst equation formula is we're going to say Ecell. Ecell here represents our nonstandard cell potential and it equals our standard cell potential which is E° minus 0.05916N times log of Q. N equals the moles of electrons that are transferred between our species within our redox reaction and Q is just our reaction quotient.

So here this represents our Nernst equation formula. Again, it helps us to determine our nonstandard cell potentials when ion concentrations are not equal to 1 molar.

The Nernst Equation Example

Here it says to calculate the cell potential for reaction to 25°C when given the following ionic concentrations and standard reduction potentials. Here we're given the overall redox reaction. As this we're given the concentrations of our ions of cobalt 3 ion and magnesium ions as one molar and 0.0033 molar respectively. In addition to this, we're given the standard reduction potentials in the form of these half reactions.

Now notice they ask us to determine cell potential, not standard cell potential. There is a difference. So to find our nonstandard cell potential we use the Nertz equation. So that's going to be cell potential, which is Ecell equals standard cell potential, which is E°cell minus 0.05916n times log of Q. Here we're going to find out what Q is. First, remember, Q equals products over reactants. It ignores solids and liquids, so here it would equal Mg^2+3 because of the three divided by Co^3+2 because of the two.

Here we plug in the values. So this is 0.0033^3 divided by one molar squared. Here. That gives me a Q of 3.5937×10-8. You're not rounding until I get my final answer at the end. So we find out what Q is. Now what do we have to find next? Let's try to find out what n is. n is the number of multiple electrons transferred. Remember, the moles of electrons have to match in both half reactions. This one here has three and this one here has two. Their lowest common multiple is 6. OK, so it's six electrons that have been transferred.

This would also explain our coefficients. Here those coefficients came into play because we had to multiply this by two to give us 6 electrons and we had to multiply this by three to give us 6 electrons. Here this will be log of 3.5937×10-8. Now remember multiplying your half reactions does nothing to our standard reduction potentials. They state those numbers because those values are based on the identity of the elements being reduced, not on the amount of them. So I can multiply these by a million. These reduction potential would stay the same.

So we found out almost everything. The only piece that's missing is our standard is our standard cell potential. Remember standard cell potential, which is E°cell equals cathode minus anode cathode. Remember, that's the site of reduction. Anode is a site of oxidation. If we look at the overall redox reaction, that'll help us determine what's been oxidized and what's been reduced. Cobalt three has an oxidation number of plus three. Because it charges +3 and it goes to 0, its oxidation number has been reduced, therefore it is the cathode. Magnesium goes from zero to +2. Its oxidation number increased, so it's been oxidized O.

Now we can do cathode minus anode, so that would be cathode is 1.82 volts -A - 2.37 volts. Now remember, a minus of a minus really mean that we're adding them together. So that comes out to 1.82 volts plus 2.37 volts. So that comes out to 4.19 volts. That'd be our standard cell potential. So now we do 4.19 volts minus 0.059166 times log of our Q value. When you punch this into your calculator, you'll get back 4.26 volts. So this would represent our regular nonstandard cell potential. So that would be our final answer.

If [Br^–] = 0.010 M and [Al³⁺] = 0.022 M, predict whether the following reaction would proceed spontaneously as written at 25ºC:

Al (s) + Br₂ (l) ⇌ Al³⁺ (aq) + Br^– (aq)

Standard Reduction Potentials

Al³⁺ (aq) + 3 e^– → Al (s)       E°_red = –1.66 V

Br₂ (l) + 2 e^– → Br^– (aq)     E°_red = +1.09 V

• A. 2.90 V
• B. 2.60 V
• C. 3.03 V
• D. 2.75 V

Determine [Fe²⁺] for the following galvanic cell at 25ºC if given [Sn²⁺] = 0.072 M, [Fe³⁺] = 0.0219 M, and [Sn⁴⁺] = 0.00345 M. 

Sn²⁺ (aq) + 2 Fe³⁺ (aq)  ⇌. Sn⁴⁺ (aq) + 2 Fe²⁺ (aq)     E_cell = + 0.68 V

Standard Reduction Potentials

Sn⁴⁺ (aq) + 2 e^–  →. Sn²⁺ (aq)      E°_red = + 0.151 V

Fe³⁺ (aq) + e^–  →  Fe²⁺ (aq)         E°_red = + 0.771 V

• A. 0.13150 M
• B. 0.00968 M
• C. 0.08180 M
• D. 0.01936 M

Here’s what students ask on this topic:

What is the Nernst equation?

The Nernst equation is a fundamental equation in electrochemistry that relates the reduction potential of a chemical reaction (specifically a half-cell in an electrochemical cell) to the standard electrode potential, temperature, and activities (often approximated by concentrations) of the chemical species undergoing reduction and oxidation. It was named after the German physical chemist Walther Nernst.

The equation is written as:

$E=E_{°}-\frac{RT}{nF}ln\left(Q\right)$

where:

• E is the cell potential (electrode potential) at non-standard conditions,
• E° is the standard electrode potential,
• R is the universal gas constant (8.314 J/(mol*K)),
• T is the temperature in Kelvin,
• n is the number of moles of electrons exchanged in the electrochemical reaction,
• F is the Faraday constant (approximately 96485 C/mol), and
• Q is the reaction quotient, which is the ratio of the activities (or concentrations) of the products to the reactants, raised to the power of their stoichiometric coefficients.

The Nernst equation allows you to calculate the potential of an electrochemical cell under different conditions, and it helps to understand how changes in concentration or temperature can affect the voltage of a battery or the efficiency of an electrochemical process.

How can the Nernst equation be used?

The Nernst equation is a fundamental equation in electrochemistry that relates the reduction potential of a chemical reaction to the standard electrode potential, temperature, and the activities (or concentrations) of the reactants and products. It is given by the formula:

$E=E_{°}-\left(\frac{RT}{nF}\right)*ln\left(Q\right)$

where E is the cell potential, E^° is the standard electrode potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons exchanged in the electrochemical reaction, F is the Faraday constant, and Q is the reaction quotient.

You can use the Nernst equation to:

1. Calculate the cell potential of an electrochemical cell under non-standard conditions.
2. Determine the equilibrium constant for a redox reaction at a given temperature.
3. Predict the direction of an electrochemical reaction by comparing the cell potential to zero.
4. Estimate the concentration of ions in a solution when given the cell potential and other ion concentrations.
5. Understand the effects of temperature and concentration on the voltage of an electrochemical cell.

This equation is crucial for understanding and predicting the behavior of batteries, fuel cells, and natural biochemical processes involving redox reactions.

When do we use the Nernst equation?

The Nernst equation is used to determine the electric potential of a cell under non-standard conditions. In other words, it's applied in electrochemistry to calculate the reduction potential of an electrochemical cell when the concentrations of ions participating in the redox reaction are not at standard conditions, which typically means 1 molar concentration and 1 atmosphere of pressure for gases.

You'll use the Nernst equation when you want to find out the potential of a half-cell or full cell when temperature, ion concentration, or pressure deviates from standard-state conditions. It's particularly useful for understanding how the potential will change with varying concentrations of reactants and products, which is essential in predicting the direction of the reaction and its equilibrium position. The equation is also instrumental in determining the pH of a solution, as it relates to the hydrogen ion concentration.