3.4. A 2.3 L cylinder containing nitrogen gas at a pressure of 2.8 atm is connected to a 5.5 L
r containing nitrogen at 17.3 atm. What is the final pressure when both the cylinders have achieved equilibrium (reached the same pressure)?
6. An analytical procedure requires a solution of chloride ions. How many grams of BaCl2 must be dissolved to make 360 ml of 0.2 M Cl– ?
(M BaCl2 = 208 g/mol)
7. Find the concentration of chloride ions when 344.4 mL of 2.4 M NaCl is mixed with 364 mL of 2.9 M KCl?
8. A sample of an unknown gas had a density of 1.45 g/L at 20.5 °C and 1.2 atm. Calculate the molar mass of the gas.
(R = 0.08206 L·atm·mol-1·K-1)
9.(a) A 20.0 L container at 303 K holds a mixture of two gases with a total pressure of 5.00
here are 2.00 mol of Gas A in the mixture, how many moles of Gas B are present? (R = 0.0821 L • atm/(K • mol))
(b) The gas in a 250. mL piston experiences a change in pressure from 1.00 atm to 2.80 atm. What is the new volume (in mL) assuming the moles of gas and temperature are held constant?
(c) Small quantities of Oxygen can be produced by the decomposition of mercury(II) oxide as shown below. Typically, the oxygen gas is bubbled through water for collection and becomes saturated with water vapor. Atomic weight of HgO = 216.6 amu, Atomic weight of Oxygen = 32.00 amu)
2 HgO(s) → 2 Hg(ℓ) + O₂(g)
(i) Assuming that 3.05 grams of HgO was used in this reaction, determine the number of moles of oxygen gas formed.(According to the above chemical equation)
(ii) Assuming 310. 0 mL of Oxygen gas was collected at at 29°C, calculate the pressure of the Oxygen gas that was collected. (R = 0.0821 L • atm/(K • mol)
(iii) If the vapor pressure of water at this temperature equals to 0.042 atm, calculate the pressure reading of this experiment.
10.Question 1: What is a player’s « reaction function » in a Bertrand game ?
Question 2: What is a subgame
subgame perfect Nash equilibrium?
Question 3: In which situations should we need the mixed extension of a game?
Question 4: Find, if any, all Nash equilibria of the following famous matrix game:
U (2,0) (3,3)
D (3,4) (1,2)
Question 5: What is the difference between a separating equilibrium and a pooling equilibrium
in Bayesian games?
Question 6: Give another name for, if it exists, the intersection of the players’ best-response
« functions » in a game?
Question 7: assuming we only deal with pure strategies, the Prisoner’s Dilemma is a situation
No Nash equilibrium One sub-optimal Nash equilibrium
One sub-optimal dominant profile No dominant profile
Question 8: If it exists, a pure Nash equilibrium is always a profile of dominant strategies:
Question 9: All games have at least one pure strategy Nash equilibrium:
Question 10: If a tree game has a backward induction equilibrium then it must also be a Nash
equilibrium of all of its subgames:
Question 11: The mixed Nash equilibrium payoffs are always strictly smaller than the pure
Nash equilibrium payoffs:
Question 12: Which of the following statements about dominant/dominated strategies is/are
I. A dominant strategy dominates a dominated strategy in 2x2 games.
II. A dominated strategy must be dominated by a dominant strategy in all games.
III. A profile of dominant strategies must be a pure strategy Nash equilibrium.
IV. A dominated strategy must be dominated by a dominant strategy in 2x2 games.
I, II and IV only I, II and III only II and III only
I and IV only I, III and IV only I and II only
Question 13: A pure strategy Nash equilibrium is a special case of a mixed strategy Nash
Question 14: Consider the following 2x2 matrix game:
U (3,2) (2,4)
D (-1,4) (4,3)
The number of pure and mixed Nash equilibria in the above game is:
Exercise (corresponding to questions 15 to 20 below): assume a medical doctor (M)
prescribes either drug A or drug B to a patient (P), who complies (C) or not (NC) with each of
this treatment. In case of compliance, controlled by an authority in charge of health services
quality, the physician is rewarded at a level of 1 for drug A and 2 for drug B. In case of noncompliance, the physician is « punished » at -1 level for non-compliance of the patient with
drug A and at -2 level for non-compliance with drug B. As for the compliant patient, drug A
should give him back 2 years of life saved and drug B, only 1 year of life saved. When noncompliant with drug A, the same patient wins 3 years of life (due to avoiding unexpected
allergic shock for instance), and when non-compliant with drug B, the patient loses 3 years of
Question 15: You will draw the corresponding matrix of the simultaneous doctor-patient game.
Question 16: Find, if any, the profile(s) of dominant strategies of this game.
Question 17: Find, if any, the pure strategy Nash equilibrium/equilibria of this game.
Question 18: Find, if any, the mixed strategy Nash equilibrium/equilibria of this game.
Questions 19 and 20: Now the doctor prescribes first, then the patient complies or not: draw
the corresponding extensive-form game (= question 19) AND find the subgame perfect Nash
14.A mathematician diagnosed with schizophrenia is fooling himself by playing
with a list L containing n distinct integers in just his
ng n distinct integers in just his thoughts.
He plays turns on the list. In each turn he does the following–
He takes a number (in its index’s order ) and swap it with any number in
the list including itself i.e. if it swap it with itself it doesn’t move at all (The
selection of the number is completely random).
He does the same for all the elements in their index’s order in that turn.
If initially the list was unsorted, such that, no element was in sorted position,
then find the probability that the list is sorted after m such turns.
Note : Take the assumption that if an element is not in its sorted
position then it can be in any other n − 1 positions equally likely.
16.A uniform beam of length L
and mass m shown in Figure
P12.16 is inclined at an angle
u to the horizontal. Its
izontal. Its upper
end is connected to a wall by
a rope, and its lower end rests
on a rough, horizontal sur-
face. The coefficient of static
friction between the beam
and surface is ms. Assume
the angle u is such that the static friction force is at its
maximum value. (a) Draw a force diagram for the beam.
(b) Using the condition of rotational equilibrium,
find an expression for the tension T in the rope in
terms of m, g, and u. (c) Using the condition of trans-
lational equilibrium, find a second expression for T in
terms of ms, m, and g. (d) Using the results from parts
(a) through (c), obtain an expression for ms
vertical component of this force. Now solve the same
problem from the force diagram from part (a) by com-
puting torques around the junction between the cable
and the beam at the right-hand end of the beam. Find
(e) the vertical component of the force exerted by the
pole on the beam, (f) the tension in the cable, and
(g) the horizontal component of the force exerted
by the pole on the beam. (h) Compare the solution
to parts (b) through (d) with the solution to parts
(e) through (g). Is either solution more accurate?
19. Sir Lost-a-Lot dons his armor and sets out from the
castle on his trusty steed (Fig. P12.19). Usually, the
drawbridge is lowered to a horizontal position so that
the end of the bridge rests on the stone ledge. Unfor-
squire didn’t lower the draw-
ing only the angle u. (e) What happens if the ladder
is lifted upward and its base is placed back on the
ground slightly to the left of its position in Figure
17.5 (5 points) There is H3PO4 available in stock solution that has an assay of 77% and a density
cific gravity) of 1.24. What is the molarity of this solution?
Using this solution how would 1 L of 2N solution be made.
6 (5 points) H3BO3 is needed to neutralize 20 ML of a 2N solution of NaOH. How much of the acid should I put in 40 ML of the water to exactly neutralize this solution?
7 (6 points) Describe how to make the solutions below :
20% w/v Salt in water.
20% v/v alcohol in water
20% w/w NaCl in water.
8 (three points) I have 0.6 g/dl solution of NaOH. What is M? Whan is N?
9 (six points) There are 3000mL of 3M NaOH. How much of the following do I need to neutralize? (watch your M’s and N’s
a) 3M H3PO4
b) 2M H2SO4
c) 1M HCL
10 (20 points) The following solutions of NaOH are mixed together 20ML of 3N, 40mL of 2N, 60mL of 1N, 80 mL of 4N, and 100mL of 5N.
a) What is the volume and normality of the final solution?
b) How much 4M sulfuric acid would I need to neutralize?
c) How much stock solution of sulfuric acid with an assay of 77% and a specific gravity of 1.14 would I need?
d) How many grams of HCl would I have to put in a 300mL solution of HCl in water to neutralize?
11 (3 points) How much 5N solution can I make with 98 grams of H3PO4 ?
12 (5 points) How much 5N solution of H3AsO4 can I make with 57 mL of stock solution that is 84% assay and 1.14 specific gravity?
13 (5 points) If we have a 4N solution of HCl that has 0.03645g of HCl in the solution, how many microliters of solution do we have?
14 (10 points) If we have 66mL of a solution of concentrated NaOH that has an assay of 88% and a specific gravity of 1.24, how much 3N H3AsO4 can be neutralized?
15 (ten points) If I have 17 mL of a 20% w/v solution of NaOH and I want to neutralize with H2SO4 that is available in a 4% w/v solution, how much of this solution will be required.
16 (ten points)
a) I have an 12mL of Ba(Cl)2 that is 78% assay that contains 8 grams of Ba(Cl)2. What is the specific gravity?
b) How many Moles of Ba(Cl)2 are there?
c) If I have 80 grams of NaOH in a liter of solution that is of an unknown specific gravity, can I calculate molarity and what is it?
d) What is the difference between molarity and normality?
e) I have 77 ml of 77% salt in water. How much 11% can I make?
18.I have the following budget constraints for an agent. In the first period of his life, he only can get
et loans (he doesn't "earn" income). With the loans (L) he needs to decide between first period consumption (C1) and investment (I). The amount invested will allow him to get a second period income Y with probability P which is increasing in I (therefore, P(I)), In case of success and the person obtain Y, the individual should use Y to repay the loan (L) that he requested in the first period and consume in the second period (C2). However, with probability 1 - P(I), the person don't get Y and therefore only consume C1. Note that if the individual only invest the loan (L=I) and don't obtain Y, he can't consume anything. That motivates him not to invest the whole loan and keep part of the loan in order to warrant at least first period consumption. Therefore, considering B the parameter for the time preference, the problem would be:
max U=ln(C1) + Bln(C2)
s.t: L = I + C1
Y(I) = L(1+r) + C2
with Probability P(I)
s.t: L = I + C1
with probability 1 - P(I)
My question is, Have you ever seen something like this? If yes, how to proceed? What is more important, I really need a bibliography (a book or article talking about this)