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.