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)
