Plotting Carnot Engine Cycle on PV Diagram

[Sorelle]

Homework Statement

I am tasked to create a PV Diagram of a Carnot Engine Cycle. I must find pressure, volume, Q, W, ΔU, and ΔS on all four points. This is what has been given to me by my teacher:

a to b : Isothermal
b to c: Adiabatic
c to d: Isothermal
d to a: Adiabatic

T_C = 300 K
T_H = 1700 K
p_c = 1.01*10⁵ Pa
v_c = 0.01 m³
Q_{a to b} = 300 K
γ (gamma) = 1.40

Homework Equations

(1) p₁v₁=p₂v₂
(2) W=nRT ln(V₂/V₁)
(3) p₁v₁^γ=p₂v₂^γ
(4) W= p₁v₁=p₂v₂ / (γ-1)
(5) T₁V₁^(γ-1) = T₂V₂^(γ-1)
(6) W = nRT ln(V₂/V₁)
(7) pv = nRT

The Attempt at a Solution

Using the above equations I managed to get point b's pressure and volume. What I got for point b:
v_b = 0.028 m³
p_b = 2.389*10⁴ Pa
First, I used the gas law equation (7) to get moles. This came out to n = 0.405 moles. Then, I used equation (5) to get volume, solving for V_b^(y-1). I then used equation (3) to get pressure, solving for p_b. I then got W = -852.7 J for the path b to c using equation (4). This seems kind of odd to me and I'm not sure if its correct because it's doing more work than the amount of heat it is providing. I assume Q_{a to b} = is Q_H ? It seems so low though.

I'm trying to figure out how am I going to get points d and a without knowing pressure or volume on those points. A class mate had suggested I used equation (5) for a to b and c to d, but those paths are isothermal. Isn't equation (5) adiabatic only? I don't know if I can even use the gas law because I'd need pressure or volume and I don't have that.

There is also the idea that I have to use equation (6) and solve for volume that way. The problem is I don't know where to start. I'm a little rusty on my calculus (it's been about 4 years). I tried to break it down to W = nRT(ln (V2) - ln (V1))= nRT( 1/V2 - 1/V1), but this doesn't seem to help and I may have done it wrong.

I'd be grateful for any kind of help.

 

[KataKoniK]

Thank you for your post. It seems like you have made some progress in solving the problem, but you still have some questions and uncertainties. Let me try to provide some guidance and clarification.

Firstly, let's review the Carnot engine cycle and its key principles. A Carnot engine is a theoretical engine that operates between two heat reservoirs at different temperatures (TC and TH). It follows a specific cycle consisting of four reversible processes: two isothermal processes and two adiabatic processes. The efficiency of a Carnot engine is given by the Carnot efficiency equation: η = (TH - TC)/TH. In order to calculate the efficiency, we need to know the temperatures of the two reservoirs.

Now, let's address your specific questions and concerns:

1. Qa to b = 300 K: This is not the heat provided to the system, but rather the temperature at point a. In order to calculate the heat provided to the system, we need to know the temperature difference between the two reservoirs (TH and TC). Without this information, we cannot calculate Qa to b.

2. W = -852.7 J for the path b to c: This seems correct. Remember that in an adiabatic process, there is no heat exchange (Q=0), so all the work done is due to the change in internal energy (ΔU = W). Also, keep in mind that work is done by the system when the volume decreases (negative work).

3. Points d and a: In order to calculate the properties at points d and a, we need to use the equations for the isothermal processes (a to b and c to d). These equations are different from the adiabatic equations, as you correctly pointed out. For isothermal processes, we use the ideal gas law (pv = nRT) and the equation W = nRT ln(V2/V1). In order to solve for the unknown variables at points d and a, we need to use the information given in the problem (TC, TH, pc, vc) and the fact that the two isothermal processes are reversible (which means that the pressure and volume remain constant throughout the process).

4. Equation (6): This equation is correct, but you need to use it for the isothermal processes, not the adiabatic ones. Also, keep in mind that you need to use the natural logarithm