10 kg of gaseous oxygen is compressed in a piston-cylinder device from an initial state of 0.8 m^3/kg, 25 C to a final state of 0.1 m^3/kg, 287 C, with all heat transfer taking place with the surroundings at 25 C. If the process is reversible, determine:

a) the work done in the process;

b) the overall heat transfer to the oxygen; and

c) the change in entropy of the oxygen and of the surroundings

I struggle a lot with thermodynamics, could you please show all work and if possible state the names of equations used. I will rate! Thank you.

Air undergoes a polytropic process in a piston–cylinder assembly from p₁ = 1 bar, T₁ = 295 K to p₂ = 9 bar. The air is modeled as an ideal gas and kinetic and potential energy effects are negligible.

For a polytropic exponent of 1.2, determine the work and heat transfer, each in kJ per kg of air,
(1) assuming constant c_v evaluated at 300 K.
(2) assuming variable specific heats.

W/m = -187.211 kJ/kg

Q/m= -93.563 kJ/kG

Determine the heat transfer, in kJ per kg of air, assuming variable specific heats.

Let the number of tropical cyclones per year in the Atlantic Basin (?) have a discrete uniform between 5 to 15; i.e. ? ∈ {5, 6, … , 15} with ?_X(?) = 1 /11. If each cyclone can turn into a major hurricane with probability 0.3; find the expected number of major hurricanes per year (?). Hint: Use the Law of Interated Expectation; ?[?] = ?[?[?|?]].

A machine prints a word and the number of letters in this word is a Poisson distributed random variable with parameter λ (so it could possibly have zero letters). However, each letter in the word is printed incorrectly with probability 2/3 independently of all other letters. Compute the expectation and the variance of the number of incorrect letters in the word that the machine prints.

Four cards bearing the numbers 2, 3, 4 and 5 are placed on the table. Two cards are selected from these four cards to form a two-digit number. List the sample space. Find the probability that the number formed

a) Is divisible by 3.

b) Is greater than 33

c) Is a multiple of 11

d) Is less than 55.

Suppose that, in an urban population, 50% of people have no exposure to a certain disease, 40% are asymptomatic carriers, and 10% have the disease with symptoms. In a sample of 20 people, we are interested in the probability that the sample contains a number of asymptomatic carriers that is exactly thrice the number that have the disease with symptoms. While the final numerical answer is important, the setup of the solution is moreso.

A manufacturing lot contains 40 items. It is known that 6 items are defective. A quality assurance engineer selects a random sample of 10 items and checks each to see if it is defective.

i. What is the mean and standard deviation of the number of defective items that she will sample. [4]

ii. What is the probability that she observes two or fewer defective items.

A road surface is being inspected for potholes. The number of potholes per kilometre is distributed as a Poisson random variable with rate parameter λ = 6.

i. What is the probability of not observing any potholes in a kilometre of road? [4]

ii. Suppose that an engineer inspects separate kilometre stretches of road until he observes one that contains potholes. What is the expected number of kilometre stretches of road that he will need to inspect before he observes one with potholes? [3]

iii. Suppose that 10 separate kilometre stretches of road are sampled at random. What is the expected value and standard deviation of the total number of potholes observed on the sampled roads?

List the characteristics of a multinomial experiment (select all that apply):

a) The probability that the outcome of a single trial falls into a particular category remains constant from trial to trial.

b) The trials are independent.

c) The experiment consists of n identical trials.

d) The outcome of each trial falls into one of k categories.

e) Each trial results in one of only two possible outcomes.

f) The experiment contains M successes and N − M failures.

g) The probability that the outcome of a single trial falls between two categories is equal to the area under the curve between those categories.

h) The experimenter counts the observed number of outcomes in each category.

i) The number of successes is evenly distributed over all k categories.

j) Its mean is 0 and its standard deviation is 1.

k) We are interested in x, the number of events that occur in a period of time or space.

l) We are interested in x, the number of successes observed during the n trials.

4. Emily likes bird watching. Every year she takes a vacation to a park famous for

its rare birds. She goes there for 10 days. From her past experience, she knows that on

average she can get 6 good sightings a day. A very good day for her is a day with at least

10 good sightings. Assume Poisson distribution of the number of good sightings on any day

(independently of other days).

a) What is the probability that she can get at least one very good day this time?

b) What is the expected number of very good days during this vacation?

c) What is the expected number of days she has to go bird watching in this park before

getting one very good day?

d) Extra credit: What should the average number of good sightings per day be so that

the probability that she gets at least one very good day during this vacation be at least 0.9?