Extra Credit 40
- Page ID
- 82851
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Question 17.5.8
Using the information thus far in this chapter, explain why battery-powered electronics perform poorly in low temperatures.
Solution:
Step 1: What is a battery?
A battery is a combination of electrochemical cells that produces an electric current at a constant voltage. Any galvanic cell is considered a battery. There are two types of batteries, which are primary and secondary. In short, primary batteries are single-used non-rechargeable batteries and secondary batteries are rechargeable batteries.
Step 2: Why do battery-powered electrons perform poorly at low temperatures?
According to the Nernst equation:
\(E_{cell}=E^{\circ}_{cell}-\left ( \frac{RT}{nF} \right)ln\left( Q \right )\)
\(E^{\circ}_{cell}\) \(\propto\) temperature-- which means that as the temperature increases \(E^{\circ}_{cell}\) increases and as \(E^{\circ}_{cell}\) decreases, the temperature also decreases.
Question 12.3.3
Tripling the concentration of a reactant increases the rate of a reaction nine times. With this knowledge, answer the following questions:
- What is the order of the reaction with respect to that reactant?
- Increasing the concentration of a reactant by a factor of four increases the rate of a reaction four times. What is the order of the reaction with respect to that reactant?
Solution:
#1: What is the order of the reaction with respect to that reactant?
The general equation for the reaction rate is :
rate = k [A]m
where k is the rate constant, [A] is the concentration of the reactant, and m is the order of the reaction. The order of the reaction is the term in which the concentration is raised to. If tripling the concentration of a reactant increases the rate of a reaction nine times the reaction rate is as follows:
9 · rate = k [3A]m
where 9 represents the magnitude of the rate of the reaction, [3A] is the concentration times three, and m is the unknown order of the reaction that needs to be solved for.
9 · k[A]m = k[3A]m ⇒ cancel the common terms
9 = 3m
32 = 3m
take the log of both sides and cancel the common terms,
m = 2 , so the order of the reaction with respect to reactant is 2.
#2: Increasing the concentration of a reactant by a factor of four increases the rate of a reaction four times. What is the order of the reaction with respect to that reactant?
The rate of the reaction is as follows:
4 · rate = k[4A]m
Question 12.5.12
In terms of collision theory, to which of the following is the rate of a chemical reaction proportional?
- the change in free energy per second
- the change in temperature per second
- the number of collisions per second
- the number of product molecules
Solution:
The Collision theory is a set of principles that states that reacting particles can form products when they collide with one another if the collisions have enough kinetic energy and the correct orientation. Particles that lacking kinetic energy may collide, but the particles will only bounce off one another unchanged. An unsuccessful collision (a) will result in no product formation while a successful collision (b) will form a product.
In order for a collision to be successful:
Requirement 1: Molecules must Collide to React - In order to react, molecules must collide. If molecules A and B are to react, they must disrupt some of their existing bonds to create new ones needed in the products. This is called collision. The frequency of collision in gas of A and B particles is proportional, so if A is doubled, then A and B will both be doubled. So the Rate = k[A][B].
- Low concentration = Few collisions
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High concentration = More collisions
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Requirement 2: Sufficiently Energetic: Reacting particles can form products when they collide with one another provided those collisions have enough kinetic energy and the correct orientation. Particles that lack the necessary kinetic energy may collide, but the particles will simply bounce off one another unchanged. A reaction will not take place unless the particles collide with minimum energy called activation energy. The activation energy is the minimum energy required to make a reaction occur. The kinetic energy of a gas is directly proportional to temperature. As temperature increases, molecules gain energy and move faster and faster. Therefore, the greater the temperature, the higher the probability that molecules will move with the necessary activation energy for a reaction to occur.
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Requirement 3: Sufficiently Oriented: Not every collision is a guarantee success because molecules also need to collide with the right orientation. Liquid and gas molecules are in constant motion and can collide in different ways for them to react.
- In terms of the collision theory, the rate of the chemical reaction is proportionate to
- (2) The change in temperature per second and (3) the number of collisions per second.
Question 21.4.7
Which of the following nuclei is most likely to decay by positron emission? Explain your choice.
- chromium-53
- manganese-51
- iron-59
Solution:
What is a positron?
A positron is a particle of matter that is similar to an electron in that their masses are equal but have opposite charge. It has a +1 charge and is represented as
\(_{+1}^{0}\textrm{ß}\). Positron emission takes place when the neutron to proton ratio is low. In order to determine if a nuclei is most likely to be a positron decay we must calculate the number of neutrons and protons to identify the ratio.
1. Chromium-53:
Number or neutrons = Mass - Number of protons
Neutrons = 53 - 24
Neutrons = 29
The neutron to proton ratio for chromium-53 is
Ratio = \(\frac{29}{24}\) = 1.21
2. Manganese-51:
Number or neutrons = Mass - Number of protons
Neutrons = 51 - 25
Neutrons = 26
The neutron to proton ratio for manganese-51 is
Ratio = \(\frac{26}{25}\) = 1.04
3. Iron-59:
Number or neutrons = Mass - Number of protons
Neutrons = 59 - 26
Neutrons = 33
The neutron to proton ratio for chromium-53 is
Ratio = \(\frac{33}{26}\) = 1.27
Since Manganese-51 has the lowest neutron to proton ratio, it is the most likely to decay by positron emission.
Question 20.2.11
Using the activity series, predict what happens in each situation. If a reaction occurs, write the net ionic equation; then write the complete ionic equation for the reaction.
- Platinum wire is dipped in hydrochloric acid.
- Manganese metal is added to a solution of iron(II) chloride.
- Tin is heated with steam.
- Hydrogen gas is bubbled through a solution of lead(II) nitrate.
Solution:
The activity series is helpful in determining if a reaction will occur. The most reactive elements are placed at the top while the least reactive elements are placed at the bottom. A metal can only displace metal ions listed below it in the activity series, but not above.
1. Platinum is one of the least reactive elements and on the activity series and will not react with the hydrogen ions. Therefore, there is no reaction.
2. A reaction will occur because manganese is higher than iron on the activity series and will displace the iron ions.
- Net Ionic Equation: Mn(s) + 2Cl(aq) → MnCl2(s)
- Complete Ionic Equation: Mn(s) + FeCl2(aq) → MnCl2(s) + Fe2+(aq)
3. A reaction will occur because tin is more reactive than hydrogen and will result in the displacement of hydrogen.
- Net Ionic Equation: Sn(s) + 2O(aq)→ SnO2(aq)
- Complete Ionic Equation: Sn(s) + 2H2O(l) → SnO2(aq) + 2H2(g)
4. A reaction will occur because lead is placed higher on the activity series.
- Net Ionic Equation: H2(g) + 2NO3(aq) → 2HNO3(aq)
- Complete Ionic Equation: H2(g) + Pb(NO3)2(aq) → 2HNO3(aq) + Pb2+(aq)
Question 20.5.6
Although the sum of two half-reactions gives another half-reaction, the sum of the potentials of the two half-reactions cannot be used to obtain the potential of the net half-reaction. Why? When does the sum of two half-reactions correspond to the overall reaction? Why?
Solution:
The sum of two half-reactions cannot be used to obtain the potential of the net half-reactions since cell potentials are not a state function. A state function is a function in which the value is not dependent on the path taken to reach it.
The sum of the two half-reactions correspond to the overall reaction when the reactions are under standard conditions.
It can be solved by using the equation: \(E^{\circ}_{(overall)}=E^{\circ}_{(cathode)}+E^{\circ}_{(anode)}\)
Question 24.6.2
In CFT, what causes degenerate sets of d orbitals to split into different energy levels? What is this splitting called? On what does the magnitude of the splitting depend?
Solution:
The Crystal Field Theory (CFT) is a bonding theory that describes the degeneration of electron d-orbitals due to the presence of ligands. It also describes the strength of the metal-ligand bonds which can cause the energy of the system to change. The shape and occupation of these d-orbitals are important in giving an accurate description of the bond energy and properties of the transition metal.
- When ligands approach the metal ion, more opposition can occur from the d-orbital electrons than others, based on the geometric structure of the molecule. Because ligands come in from different directions, the d-orbitals interact differently.
- The splitting is called a different name based on the geometric structure of the molecule. In an octahedral structure, the splitting is called \(\Delta_{o}\). Likewise, in a tetrahedral complex, the splitting is called \(\Delta_{t}\).
- The magnitude of the splitting depends on the charge on the metal ion, the position of the metal in the periodic table, and the nature of the ligands.
- For additional readings, check out 24.6: Crystal Field Theory.