Answer:

A) Q1 = (3/2)P1V1[A - 1]

B) W2 = P1V1(In A)

C) W3 = P1V1(1 - A)

Explanation:

A) From first law of thermodynamics and applying to the question, we have;

ΔU = Q - W

Where,

ΔU = change in internal energy

Q = the heat absorbed

W = the work done

Now, because the first process occurs at constant volume, the work done is zero:

Thus,

ΔU = Q - 0

ΔU = Q

The change in internal energy is given by;

ΔU = nCvΔt

where;

n = the number of moles of the gas

R = the gas constant,

Cv = the specific heat at constant volume

Δt = The change in temperature i.e T2 - T1.

Now, using the ideal gas law, let us find an expression for n and Δt

P1V1 = nRT1

n = P1V1/RT1

T1 = P1V1/nR

Now, the specific heat at constant volume is Cv = (3/2)R

Now, from the question, since it's pressure has reached AP1, we can calculate the temperature T2 by using the ideal gas law at the new conditions of the gas as;

AP1V1 = nRT2

T2 = AP1 V1/ nR

Now, we are to express the heat added in terms of p1, V1, and A

Q = ΔU = nCv(T2 - T1)

From earlier, we saw that,

T1 = P1V1/nR

Putting equation of T2 and T1 into the energy equation to get;

Q = nCv((AP1 V1/ nR) - P1V1/nR)

Q = Cv • P1V1/R (A - 1)

Now, from earlier, we saw that Cv = (3/2)R. Thus,

Q = (3/2)R • P1V1/R (A - 1)

Q = (3/2)P1V1[A - 1]

B) Here again, we are to express work done in step 2 in terms of p1, V1, and A.

This process is an isothermal process because temperature is constant and so work done is given as; W = nRT In(V2/V1)

T = T1 because temperature is constant

From earlier, we saw that;

n = P1V1/RT1 and

But in this process, it's

n = P1V1/RT1 and thus,

V2 = nRT2/P1

We also saw that T2 = AP1 V1/ nR

V1 = nRT2/AP1

Plugging in the relevant values into, W = nRT In(V2/V1), we obtain;

W = (P1V1/RT1) • RT1 • In((nRT2/P1)/(nRT2/AP1)

W = P1V1(In A)

C) In step 3,we have and isobaric process because the pressure is constant.

Work done in this case is given by ;

W = P(V1 - V2)

Because V2 in now the final volume while V1 is now the the initial volume

Now, P is P1 because it's an isobaric process.

From earlier, we saw that,

V1 = nRT2/AP1 and V2 = nRT2/P1

And that T2 = AP1 V1/ nR

Thus,

V1 = V1 and V2 = AV1

Thus, W = P1(V1 - AV1) = P1V1(1 - A)