Find the equation of state of each gas

AI Thread Summary
The discussion revolves around finding the equations of state for different gases using thermodynamic principles. The first gas follows the ideal gas law, while the second gas incorporates corrections for intermolecular forces and volume. Participants highlight the challenge of assuming the right-hand side of the equation as simply nRT and emphasize the need for valid relationships across various temperatures. The concept of treating the equations as functions of temperature is discussed, with clarification that these functions can differ for each gas. The conversation concludes with a suggestion to consider molar volume instead of total volume to simplify the equations.
curious_mind
Messages
48
Reaction score
9
Homework Statement
Consider three gases with ##(P_1,V_1),(P_2,V_2)## and ##(P_3,V_3)##. It is found that when the first two are in equilibrium the following condition is satisfied: ##P_1V_1 =\left(P_2 +\frac{a}{V_2^2}\right)(V_2 −b)##, while the equation satisfied when the first and the last are in equilibrium is ##P_3(V_3 −c)=P_1V_1 e^{\frac{−d}{V_3P_1V_1}}##. Find the respective equations of state and identify them.
Relevant Equations
Equation of states of gas at temperature T##f(P,V,T)=0##
The problem is from the book "The Principles of Thermodynamics" by ND Hari dass.

It looks trivial problem, but I am not able to form logical arguements for going into next step.

For example, It seems like first gas has equation of state ##PV =nRT## and second has ## \left( P_2 +\frac{a}{V_2^2} \right) (V_2 −b) = nRT ##
But I cannot straightforward assume Right hand side of equation of state to be simply ##nRT## in general right ? So what could be valid way to proceed from thermodynamical laws ?

Thanks.
 
Physics news on Phys.org
For the relationships to hold across all T, each relationship must be of the form (first expression =second expression = some function of T).
 
  • Like
Likes curious_mind
If we equate all three relations, then it will be valid only if all three gases in equilibrium, which is not required to be found. We require to find all three gases equation of state separately, at different temperatures.

Also, how can we say that it individual ##f(P,V)## is some function of ##T## ONLY?. Right hand side of equation of state might containt terms like ##\cos (TVe^P)## etc etc or something, in general - right m? Or am I missing something very fundamental?
 
Last edited:
curious_mind said:
If we equate all three relations
I did not say that. The "some function of T" does not have to be the same for each.
curious_mind said:
Also, how can we say that it individual ##f(P,V)## is some function of ##T## ONLY?.
It is the same principle as "separation of variables" in PDEs.
We know that ##T_1=g_1(P_1,V_1)## and ##T_2=g_2(P_2,V_12)## for some functions ##g_1, g_2##. So at any given temperature T we know ##g_1(P_1,V_1)=g_2(P_2,V_12)##. And these are the forms you are given.
 
  • Like
Likes curious_mind
It seems to me your original approach was correct, except I would assume the V is molar volume rather than volume itself, so you would get rid of the n's in the equations.
 
  • Like
Likes curious_mind
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
Thread 'Trying to understand the logic behind adding vectors with an angle between them'
My initial calculation was to subtract V1 from V2 to show that from the perspective of the second aircraft the first one is -300km/h. So i checked with ChatGPT and it said I cant just subtract them because I have an angle between them. So I dont understand the reasoning of it. Like why should a velocity be dependent on an angle? I was thinking about how it would look like if the planes where parallel to each other, and then how it look like if one is turning away and I dont see it. Since...
Thread 'Voltmeter readings for this circuit with switches'
TL;DR Summary: I would like to know the voltmeter readings on the two resistors separately in the picture in the following cases , When one of the keys is closed When both of them are opened (Knowing that the battery has negligible internal resistance) My thoughts for the first case , one of them must be 12 volt while the other is 0 The second case we'll I think both voltmeter readings should be 12 volt since they are both parallel to the battery and they involve the key within what the...
Back
Top