Thermo question, two methods of changing volume/temp

[Greger]

does this look right?

for the quasistatic case you can use thermodynamics to find the temperature at any time,

for the other case you have to use dU=dW and so on since its not quasistatic

 

[ehild]

does this look right?

for the quasistatic case you can use thermodynamics to find the temperature at any time,

for the other case you have to use dU=dW and so on since its not quasistatic

It is correct for the constant external pressure. But what is the final temperature in the quasi-static case? It is true that T_f=PV_f/(NK_B) but P changes during the process. How do you get the final pressure P_f?


ehild

 

[Greger]

Oh right,

so the quasistatic case should be:

T_f=P_fV_f/(NK_B)

for some reason i forgot that the pressure was changing in that case as well,

this makes more sense, in the quasistatic case you can find the state of the system at any point in terms of the state variables, and P is one of those too!

thanks ehild =]

 

[ehild]

Can you show what you got? Just to complete the solution, so as other people learn from it.

ehild

 

[Greger]

quasistatic case :

T_f=P_fV_f/(NK_B)

under constant pressue:

T_f=2P_ex(Vi-Vf)/3Nk_B + Ti

 

[ehild]

quasistatic case :

T_f=P_fV_f/(NK_B)

under constant pressue:

T_f=2P_ex(Vi-Vf)/3Nk_B + Ti

Well, it is not the solution yet. You need to give T_f in terms of the initial and final volumes and the initial temperature. Do you know the equation that governs a quasistatic adiabatic process?

ehild

 

[Greger]

Oh,

do you mean something like

where gamma is the ratio of heat capacities under constant pressure / volume?

 

[ehild]

Yes. But you can combine it with the ideal gas law to give an equation between V and T.

ehild

 

[Greger]

yea theres afew like

TV^(gamma - 1)=constant

PV^gamma = const

but wouldnt mucking around with these and introducing them into my problem make it like

Tf=PfVf/(NKB)=P_iV_f^1-λVi^λ/(NK_B)

using λ as gamma

PV^gamma = const = C say



Tf=CV_f^1-λ/NK_B

then i have this unknown constant C

 

[ehild]

yea theres afew like

TV^(gamma - 1)=constant

That is a good start. You can find that constant as you know the initial volume and temperature:
T_fV_f^γ-1=T_iV_i^γ-1.
From the equation for the internal energy you can see that it is a mono-atomic gas, so you know Cv, and you also know the relation between Cv and Cp. What is γ-1 then?

ehild

 

[Greger]

C=NKb Tf Vf^(γ-1)=NKb TiVi^(γ-1)


Tf= Ti(Vf Vi)^(γ-1)

for γ=Cp/Cv

Tf= Ti(Vf Vi)^(Cp/Cv-1)

is that kind of what you mean?

 

[ehild]

C=NKb Tf Vf^(γ-1)=NKb TiVi^(γ-1)


Tf= Ti(Vf Vi)^(γ-1)

for γ=Cp/Cv

Tf= Ti(Vf Vi)^(Cp/Cv-1)

is that kind of what you mean?

Yes, but you made a little error:

T_f=T_i(V_i/V_f)^γ-1.
And you know the numerical value of Cp/Cv what is it?

ehild

 

[Greger]

oh whoops,


T_f=T_i(V_i/V_f)^5/3-1.



T_f=T_i(V_i/V_f)^2/3.

thanks ehild

 

[ehild]

It is the SOLUTION now

ehild