Gas Homework: Adiabatic/Isochoric/Isothermal Process

In summary, a 26-L sample of an ideal gas with (gamma)= 1.67 undergoes a series of processes, including an adiabatic compression, isochoric cooling, and isothermal expansion. The total work done on the system is found to be -69 J, but there may be a mistake due to rounding in the intermediate values used.
  • #1
MeliMo
4
0

Homework Statement


A 26-L sample of an ideal gas with (gamma)= 1.67 is at 250 K and 50 kPa . The gas is compressed adiabatically until its pressure triples, then cooled at constant volume back to 250 K, and finally allowed to expand isothermally to its original state.



Homework Equations


I've called the three states a,b, and c, so the subscripts are W_(from-to)

W_ab= (P_a*V_a - P_b*V_b)/(gamma-1)

P_b=(50 kPa)(3)=150 kPa

P_a(V_a)^gamma=P_b(V_b)^gamma

W_bc= 0

W_ca= nRT*ln(V_a/V_c)

pV=nRT
V_c=V_b


The Attempt at a Solution



The first process is adiabatic, so I used the first equation to find the work. I first needed to find the V_b, so I used the second equation: (50 kPa)(26 L)^1.67=(150 kPa)(V_b)^1.67
I found that V_b= 13 L (based on another part of the question that I solved correctly I know that they rounded it to 13).
Then W_ab= [(50)(26 )-(150)(13)]/(1.67-1) = -970 J

The second process was isochoric, so W= 0.

The third process is isothermal, so I used:
W_ca= p_a(V_a)ln(V_a/V_c) = (50)(26) * ln (26/13) = 901 J

So then the total work done is: W= 901-970 = -69 J

can someone please tell me where my mistake is? Thanks!
 
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  • #2
Check the sign. Since this is work done on the system.

Edit: I did remove the part about kPa * L = J when I quickly realized I was wrong.
 
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  • #3
Actually kPa * L = J is true...

It seems like your math is correct, but I wonder about the rounding. You're not supposed to round intermediate values like the 13L; rounding is only applied when you write out the final answer. In this case, the difference between 13L and the unrounded value is large enough that it makes a noticeable (well, more than just "noticeable") difference in the answer.
 
  • #4
Yeah, I tried answering with both the rounded and not rounded intermediate and I noticed the difference too, but it still returned both answers as incorrect. Usually when just the sign is wrong it'll say so as well... I've been going off of an example in my book that is pretty much the same problem with different values and somehow I keep getting it wrong. In any case its good to know that its not just me making some very obvious mistake, thanks!
 

Related to Gas Homework: Adiabatic/Isochoric/Isothermal Process

What is a gas?

A gas is a state of matter that has no fixed shape or volume. It is made up of particles that are constantly moving and have a large amount of space between them.

What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no transfer of heat or matter between the system and its surroundings. This means that the internal energy of the system remains constant.

What is an isochoric process?

An isochoric process is a thermodynamic process in which the volume of the system remains constant. This means that the work done by or on the system is zero, and the change in internal energy is equal to the heat added or removed from the system.

What is an isothermal process?

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. This means that there is no change in the internal energy of the system and the heat added or removed from the system is equal to the work done by or on the system.

What are some real-life examples of these processes?

An adiabatic process can be seen in the expansion of a gas in a spray can, where the internal energy of the gas decreases as it expands without any heat transfer. An isochoric process can be seen in a pressure cooker, where the volume of the system remains constant as pressure increases. An isothermal process can be seen in a refrigerator, where the temperature inside remains constant while heat is removed from the system.

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