Calculate the pH of the cathode compartment for the following reaction given Ecell = 3.01 V when [Cr3+] = 0.15 M, [Al3+] = 0.30 M, and [Cr2O72-] = 0.55 M.
2 Al(s) + Cr2O72-(aq) + 14 H+(aq) → 2 Al3+(aq) + 2 Cr3+(aq) + 7 H2O(l)
To calculate the pH of the cathode compartment, we need to understand the reaction and determine the species present in the cathode compartment.
From the given reaction:
2 Al(s) + Cr2O72-(aq) + 14 H+(aq) → 2 Al3+(aq) + 2 Cr3+(aq) + 7 H2O(l)
We can see that Al3+(aq) and Cr3+(aq) are the species present in the cathode compartment, and we need to determine the pH.
To calculate the pH, we need to use the Nernst equation, which relates the cell potential (Ecell) to the concentration of species involved in the half-reactions.
The Nernst equation can be written as:
Ecell = E°cell - (RT/nF) * ln(Q)
Where:
- E°cell is the standard cell potential,
- R is the gas constant (8.314 J/(mol∙K)),
- T is the temperature in Kelvin,
- n is the number of electrons transferred in the balanced half-reaction,
- F is the Faraday constant (96485 C/mol), and
- Q is the reaction quotient.
Given that Ecell = 3.01 V, the standard cell potential (E°cell) is equal to this value since the concentrations are not mentioned in the reaction.
We can rearrange the Nernst equation to solve for the reaction quotient (Q):
Q = e^((E°cell - Ecell) * (nF/RT))
Now, let's determine the values needed to calculate Q:
- n: From the balanced equation, we can see that 2 electrons are transferred in the half-reaction.
- F: The Faraday constant is 96485 C/mol.
- R: The gas constant is 8.314 J/(mol∙K).
- T: The temperature is not given, so we'll assume it to be 298 K (room temperature) for this calculation.
Substituting the values, we can calculate Q:
Q = e^((0 - 3.01) * (2 * 96485 / (8.314 * 298)))
Now that we have the value of Q, let's calculate it:
Q = e^(-91370.9604)
Using a scientific calculator or an online calculator, we find that Q is approximately 1.8917 × 10^(-39867).
Next, we need to calculate the concentration of Cr3+(aq) in the cathode compartment. From the balanced equation, we can see that the mole ratio between Cr2O72-(aq) and Cr3+(aq) is 1:2.
Given [Cr2O72-] = 0.55 M, the concentration of Cr3+(aq) in the cathode compartment would be half of this value:
[Cr3+] = 0.55 M / 2 = 0.275 M
Now that we have the concentration of Cr3+(aq), we can calculate the pH using the concentration of Al3+(aq) and Cr3+(aq) in the cathode compartment.
However, it's important to note that the given information does not provide any direct relationship between pH and the concentrations of Al3+(aq) and Cr3+(aq). To determine the pH, we need additional information about the specific conditions or use additional equations or concepts related to pH calculations.
Please provide more information or specify if you are looking for any particular method to calculate the pH based on the given information.