Chapter 20: Problem 91
(a) \(\mathrm{A} \mathrm{Cr}^{3+}(a q)\) solution is electrolyzed, using a current of 7.60 \(\mathrm{A} .\) What mass of \(\mathrm{Cr}(s)\) is plated out after 2.00 days? (b) What amperage is required to plate out 0.250 mol Cr from a \(\mathrm{Cr}^{3+}\) solution in a period of 8.00 \(\mathrm{h} ?\)
Short Answer
Expert verified
Based on the given information, the mass of Cr plated out after 2.00 days is approximately \(315\,\text{g}\), and the required amperage to plate out 0.250 mol Cr in a period of 8.00 h is approximately \(11.19\,\text{A}\).
Step by step solution
01
(a) Step 1: Write the balanced half-cell reaction for plating of Cr metal
The balanced half-cell reaction for the plating of Cr from a Cr³⁺ solution involves the reduction of Cr³⁺ ions to Cr(s):
\[Cr^{3+}(aq) + 3e^- \rightarrow Cr(s).\]
02
(a) Step 2: Calculate the charge
Use the formula for charge Q (in coulombs) based on the given current (I) and time (t) in seconds:
\[Q = I \times t.\]
Given current: \(I = 7.60\,\text{A}\) and time: \(t = 2\,\text{days} = 2 \times 24 \times 3600\,\text{s},\)
calculate the charge, \(Q\),
\[Q = 7.60\,\text{A} \times 2 \times 24 \times 3600\,\text{s}.\]
03
(a) Step 3: Calculate the number of moles of electrons
Use Faraday's constant to find the number of moles of electrons involved in the process, n:
\[n = \frac{Q}{F},\]
where \(F = 96485\,\text{C/mol}\).
Calculate the number of moles of electrons using the charge obtained in the previous step.
04
(a) Step 4: Calculate the number of moles of Cr plated out
Use the stoichiometry of the balanced half-cell reaction to determine the number of moles of Cr plated out:
\[1\,\text{mol Cr} \leftrightarrow 3\,\text{mol e^-}.\]
Calculate the number of moles of Cr plated out using the number of moles of electrons obtained in the previous step.
05
(a) Step 5: Calculate the mass of Cr plated out
Use the molar mass of Cr (\(51.996\,\text{g/mol}\)) to calculate the mass of Cr plated out, m:
\[m = (\text{number of moles of Cr})\times(\text{molar mass of Cr}).\]
Calculate the mass of Cr plated out using the number of moles of Cr obtained in the previous step.
06
(b) Step 1: Calculate the charge required to plate out 0.250 mol Cr
Use the stoichiometry of the balanced half-cell reaction to determine the number of moles of electrons needed for plating 0.250 mol Cr:
\[1\,\text{mol Cr} \leftrightarrow 3\,\text{mol e^-}.\]
Calculate the number of moles of electrons needed for plating 0.250 mol Cr and then use Faraday's constant to calculate the required charge (Q).
07
(b) Step 2: Calculate the time in seconds
Convert the given time (8.00 h) to seconds:
\(t = 8.00\,\text{h} \times 3600\,\text{s/h}\).
08
(b) Step 3: Calculate the required amperage
Use the formula for charge (Q) based on current (I) and time (t) to find the required amperage I:
\[Q = I \times t.\]
Solve for the current (I) using the obtained values of charge (Q) and time (t):
\[I = \frac{Q}{t}.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Faraday's Law of Electrolysis
Faraday's law of electrolysis is a fundamental principle that describes the quantitative relationship between the amount of substance that undergoes electrolysis and the amount of electric charge that passes through the electrolyte. It states that the amount of substance deposited or dissolved at an electrode during electrolysis is directly proportional to the quantity of electric charge passed through the electrolyte.
This law can be expressed mathematically as: \[m = (Q \times M) / (n \times F)\], where \(m\) is the mass of the substance in grams, \(Q\) is the total electric charge in coulombs, \(M\) is the molar mass of the substance (in g/mol), \(n\) is the number of moles of electrons exchanged per mole of substance (the valence), and \(F\) is Faraday's constant, approximately \(96,485\) coulombs per mole of electrons.
Applying this law to an exercise means calculating the quantity of electric charge passed and using the stoichiometry of the reaction to find the amount of substance produced or consumed.
This law can be expressed mathematically as: \[m = (Q \times M) / (n \times F)\], where \(m\) is the mass of the substance in grams, \(Q\) is the total electric charge in coulombs, \(M\) is the molar mass of the substance (in g/mol), \(n\) is the number of moles of electrons exchanged per mole of substance (the valence), and \(F\) is Faraday's constant, approximately \(96,485\) coulombs per mole of electrons.
Applying this law to an exercise means calculating the quantity of electric charge passed and using the stoichiometry of the reaction to find the amount of substance produced or consumed.
Molar Mass Calculations
Molar mass calculations are integral to stoichiometry, especially in electrochemical reactions where you need to relate the weight of a substance to the number of moles. The molar mass is the weight of one mole of a substance and is expressed in grams per mole (g/mol).
To find the mass of a substance that gets deposited during electrolysis, you can use the previously calculated moles from Faraday's law of electrolysis and multiply by the molar mass of the substance: \[mass = moles \times molar\,mass\].
For instance, if you calculate the moles of chromium in an electrolysis problem, you can find the mass by multiplying the moles of chromium by its molar mass, which is \(51.996\,g/mol\) for chromium. This link between molar mass and moles is critical for accurately determining the mass of a substance in an electrochemical experiment.
To find the mass of a substance that gets deposited during electrolysis, you can use the previously calculated moles from Faraday's law of electrolysis and multiply by the molar mass of the substance: \[mass = moles \times molar\,mass\].
For instance, if you calculate the moles of chromium in an electrolysis problem, you can find the mass by multiplying the moles of chromium by its molar mass, which is \(51.996\,g/mol\) for chromium. This link between molar mass and moles is critical for accurately determining the mass of a substance in an electrochemical experiment.
Stoichiometry of Electrochemical Reactions
Stoichiometry in electrochemical reactions refers to the quantitative relationship between the elements and compounds involved in a reaction. It is based on the law of the conservation of mass, which states that mass in a closed system will remain constant, regardless of the processes that happen inside.
In electrochemistry, we often deal with reactions that involve the transfer of electrons. The stoichiometry for these reactions is important because it dictates the proportions in which substances react or are produced. For example, in the formation of chromium metal from its ions \(Cr^{3+}(aq) + 3e^- \rightarrow Cr(s)\), the stoichiometry shows that three moles of electrons are needed to deposit one mole of chromium metal.
Understanding the stoichiometry of a reaction allows you to determine the amount of electric current needed to produce a certain amount of product (or consume a given amount of reactant), which ties directly to Faraday's laws of electrolysis.
In electrochemistry, we often deal with reactions that involve the transfer of electrons. The stoichiometry for these reactions is important because it dictates the proportions in which substances react or are produced. For example, in the formation of chromium metal from its ions \(Cr^{3+}(aq) + 3e^- \rightarrow Cr(s)\), the stoichiometry shows that three moles of electrons are needed to deposit one mole of chromium metal.
Understanding the stoichiometry of a reaction allows you to determine the amount of electric current needed to produce a certain amount of product (or consume a given amount of reactant), which ties directly to Faraday's laws of electrolysis.
Current and Charge Relationship
The relationship between electric current and charge is foundational in understanding electrolysis chemistry. Electric current, measured in amperes (A), represents the flow of electrical charge, and is calculated by the rate at which charge is transferred over time.
The relationship can be formulated as: \[I = Q / t\], where \(I\) is the current in amperes, \(Q\) is the charge in coulombs, and \(t\) is the time in seconds during which the charge passes. If you want to calculate the charge that has passed during electrolysis, you can rearrange this formula to: \[Q = I \times t\].
In practical exercises, if you are given the current and the time period of electrolysis, you can use this relationship to calculate the total electric charge that has passed. This charge, in turn, is used in Faraday's law to find out the mass of the substance that has been electrolyzed. Such calculations are critical when you need to know how much current to apply to plate out a certain quantity of metal, like in the chromium plating example provided.
The relationship can be formulated as: \[I = Q / t\], where \(I\) is the current in amperes, \(Q\) is the charge in coulombs, and \(t\) is the time in seconds during which the charge passes. If you want to calculate the charge that has passed during electrolysis, you can rearrange this formula to: \[Q = I \times t\].
In practical exercises, if you are given the current and the time period of electrolysis, you can use this relationship to calculate the total electric charge that has passed. This charge, in turn, is used in Faraday's law to find out the mass of the substance that has been electrolyzed. Such calculations are critical when you need to know how much current to apply to plate out a certain quantity of metal, like in the chromium plating example provided.