pV diagram, ranking heat transfer between 4 processes

[HoboBones]

Homework Statement:

Rank the magnitude of the heat transferred with the gas in each of the four processes.

Relevant Equations:

First law of thermodynamics
Thermal energy
Ideal gas law
Work done by gas
Work in isothermal

Apologies, made a mistake when posting. Please see below post.

 

[HoboBones]

Problem and pV diagram

Problem: Rank the magnitude of the heat transferred with the gas in each of the four processes.

Given pV diagram

Attempt at solution with my questions in red

Equations used:

Thermal energy: E_th=3/2nRT
Ideal gas law: pV=nRT
Work_{by gas}=area under curve
Work in isothermal: W_{gas,isothermal}=nRTln(V_f/V_i)
First law of thermodynamics applied to gases: ΔE_th=Q-W_gas

Set up:

From ideal gas law, pV=nRT. Thus,

ΔE_th=3/2nRT=3/2pV
W_{gas, isothermal}=nRTln(V_f/V_i)=pVln(V_f/V_i)
ΔE_th=Q-W_gas --> Q=ΔE_th+W_gas

Solving for ΔE_th, W_gas, and |Q| for the processes: (I don't understand why we are solving for absolute value of Q)

I need some help solving for ΔE, W_gas, and |Q| for process 3->4

Process​	ΔEth​	Wgas​	|Q|​
1->2​	-9pV​	-6pV​	15pV​
2->3​	3/2(-pV)​	0​	1.5pV​
3->4​	0pV​	?​	?​
4->5​	3/2(pV)​	pV​	2.5pV​

Here is my attempt at process 3->4

ΔE_th3->4 = 3/2(2pV-2pV) = 0 (My professor has the answer as "0pV" and not 0, not sure why?

W_{gas,isothermal}= area under the curve?

Area of triangle + Area under rectangle = 2pV? (correct answer is -pVln(V_f/V_i)

|Q| = -pVln(2V/V) = 1.4pV (I am not sure how this is the correct answer)

Any help would be much appreciated!

 

[kuruman]

$dW=pdV.$ Replace $p$ using the ideal gas law and integrate. Note that "isothermal" means "constant $T$."

 

[HoboBones]

Update, I think I figured it out but not really understanding

For process 3->4,

W_{gas,isothermal} = area under curve = 1/2bh + A_rectangle = 2pV

We can plug in our area into the work in isothermal equation, thus

W_{gas,isothermal} = -pVln(V_f/V_i) = -2pVln(2V/V)

|Q| = ΔE_th(3->4) + W_{gas,isothermal} = 0pV -2pVln(2V/V) = |-1.386pV| = 1.4pV

 

[HoboBones]

$dW=pdV.$ Replace $p$ using the ideal gas law and integrate. Note that "isothermal" means "constant $T$."

She doesn't want us to use integrals unfortunately

 

[kuruman]

Integrals was plan A. Plan B says call the heat entering the gas during the isothermal part $Q_{34}.$ Add an extra row to the table that completes the cycle from 5 to 1 with an isochoric process. Calculate the new entries the same way you did step 2 to 3. Now add all 5 elements in each column. Note that $W_{34}=Q_{34}$ so you have one equation and one unknown, $Q_{34}.$ There is no plan C.