please help with partial differentiation

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Jitterbug
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Hi, not sure if this is in the right forum because these are first year degree qus, but if anyone can help would really appreciate it:

1) for (xdy-ydx)(x^2+y^2)
firstly had to determine that this was exact, if in the form Pdx+Qdy then
(dP/dy)=(dQ/dx)=(y^2-x^2)/((x^2+y^2)^2)

i)how do i find a function f such that df=Pdx+Qdy?
ii)what is the general solution to the equation Pdx+Qdy=0 im guessing this will be f=constant but just dont know how to get f.

2)if enthalpy of a gas is H=U+pV where U satisfies dU=TdS-pdV then what is the relationship between the variables H,S and p? how do i show that (dV/dS)with p constant is equal to (dT/dP) with s constant?

By regarding U as a function of p and V and considering 2 expressions for d2U/dpdV show that:

(dS/dV)(dT/dp)-(dS/dp)(dT/dV)=1
p v v p (these letters are constants for each differential)

thanks in advance for any suggestions x
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RichE
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(Original post by Jitterbug)
Hi, not sure if this is in the right forum because these are first year degree qus, but if anyone can help would really appreciate it:

1) for (xdy-ydx)/(x^2+y^2)
firstly had to determine that this was exact, if in the form Pdx+Qdy then
(dP/dy)=(dQ/dx)=(y^2-x^2)/((x^2+y^2)^2)

i)how do i find a function f such that df=Pdx+Qdy?
ii)what is the general solution to the equation Pdx+Qdy=0 im guessing this will be f=constant but just dont know how to get f.
You're right in two, the general soln will be f(x,y)=c where f is such that

df/dx = P and df/dy = Q.

Now we know that

df/dx = -y/(x^2+y^2).

If we integrate wrt x then we get

f(x,y) = -arctan(x/y) + g(y).

Note that we have a function g of y, instead of the usual constant, because this would disappear when we partially differentiated wrt x.

If we partial-differentiate this equation wrt y we get

df/dy = -x/(x^2+y^2) + g'(y)

which is what we want if g'(y)=0. i.e. if g is a constant.

So we can take f(x,y) = arctan(x/y) and the general solution is

arctan(x/y) = c.
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RichE
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Can you give any more information on the second one?

You have that H is a function of U,p,V and the second equation gives U in terms of T,S,p,V so I can't see that you can get H in terms of just S and p. :confused:

Or are you looking for a differential equation relating H,S,p?
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Jitterbug
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(Original post by RichE)
Can you give any more information on the second one?

You have that H is a function of U,p,V and the second equation gives U in terms of T,S,p,V so I can't see that you can get H in terms of just S and p. :confused:

Or are you looking for a differential equation relating H,S,p?

i think its just a differential equation relating H S and p. its something to do with thermodynamics and maxwell identities but not sure how to go about it.
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RichE
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But does it just involve H, s and p? How do you eliminate the other variables? Are there other conservation relating the other variables that I need to know about?
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Fermat
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(Original post by RichE)
But does it just involve H, s and p? How do you eliminate the other variables? Are there other conservation relating the other variables that I need to know about?
The gas eqn is pV = mRT (mR = const)
can you use that to eliminate T?
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nas7232
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pv = nRT

Therefore, pv = (m/M) RT

You cant eliminate T like that i dont think.

BTW i didnt read the thread i just saw the equation, this idea gas equation is in my physics syllabus.
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evariste
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(Original post by Jitterbug)
.
2)if enthalpy of a gas is H=U+pV where U satisfies dU=TdS-pdV then what is the relationship between the variables H,S and p? how do i show that (dV/dS)with p constant is equal to (dT/dP) with s constant?
H=U+pV
dH=dU+d(pV)
=TdS-pdV+vdp+pdv
=TdS+vdP.......(1)
so H=H(S,P)
let D denote partial diff
from (1)
DH/DS=T DH/Dp=v
now D^2H/DSDP=Dv/DS=D^2H/DpDS=DT/DP
which gives Dv/DS=DT/Dp.
probably same idea for second part. ill look later if i get time, got some lesson plans to write up now.
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RichE
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(Original post by evariste)
H=U+pV
dH=dU+d(pV)
=TdS-pdV+vdp+pdv
=TdS+vdP.......(1)
so H=H(S,P) (*)
let D denote partial diff
from (1)
DH/DS=T DH/Dp=v
now D^2H/DSDP=Dv/DS=D^2H/DpDS=DT/DP
which gives Dv/DS=DT/Dp.
probably same idea for second part. ill look later if i get time, got some lesson plans to write up now.
How is (*) a relationship between H,S,when T and v are also involved?
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evariste
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(Original post by RichE)
How is (*) a relationship between H,S,when T and v are also involved?
Because from (1) small changes in S and p produce small change in H. ie H varies as S and p do. it is (probably?) possible to define H in terms of other variables too.
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