(p+a/v^2)v=RT,derive the cyclic rule.i.e. show the products of the partial derivatives of p,v&T taken in a order is equal to -1?
1 Answer
DISCLAIMER: LONG AND MATH-HEAVY ANSWER!
I am pretty sure that you mean to write
#(P + a/(barV^2))barV = RT#
or
#PbarV + a/(barV) = RT#
THE CYCLIC RULE OF PARTIAL DERIVATIVES
The cyclic rule in general says:
#((delx)/(dely))_z((dely)/(delz))_x((delz)/(delx))_y = -1# .
Since
#((delbarV)/(delbarT))_P((delT)/(delP))_(barV)((delP)/(delbarV))_(T) = ?# where
#P# is implicitly a function of#T# and#barV# .
PROVING THE CYCLIC RULE USING P, V/n, T
To prove the cyclic rule, write the total derivative of
#bb(dP = ((delP)/(delT))_(barV)dT + ((delP)/(delbarV))_TdbarV)#
Now if we divide by
#cancel(((delP)/(delbarV))_P)^(0) = ((delP)/(delT))_(barV)((delT)/(delbarV))_P + ((delP)/(delbarV))_Tcancel(((delbarV)/(delbarV))_P)^(1)#
#0 = ((delP)/(delT))_(barV)((delT)/(delbarV))_P + ((delP)/(delbarV))_T#
#-((delP)/(delbarV))_T = ((delP)/(delT))_(barV)((delT)/(delbarV))_P#
Now if you recall that
#-((delbarV)/(delT))_P*((delT)/(delP))_(barV)*((delP)/(delbarV))_T = cancel(((delT)/(delP))_(barV)*((delP)/(delT))_(barV)((delbarV)/(delT))_P*((delT)/(delbarV))_P)^(1)#
Thus:
#color(blue)(((delbarV)/(delT))_P((delT)/(delP))_(barV)((delP)/(delbarV))_T = -1)#
APPLYING THE CYCLIC RULE OF PARTIAL DERIVATIVES
To use the cyclic rule for your equation of state, try rewriting it in terms of
P-form of the equation of state
#bb(P = (RT)/(barV) - a/(barV^2))#
#=> color(green)(((delP)/(delbarV))_T = -(RT)/(barV^2) + (2a)/(barV^3))#
We can use the reciprocal property of partial derivatives to not have to rewrite this in terms of
#=> color(green)(((delT)/(delP))_(barV)) = 1/((delP)/(delT))_(barV)#
#= 1/(R/(barV)) = color(green)((barV)/R)#
T-form of the equation of state
Now to rewrite in terms of
#T = 1/R[PbarV + a/(barV)]#
#=> color(green)(((delbarV)/(delT))_P) = 1/(((delT)/(delbarV))_P)#
#= 1/[P/R - a/(RbarV^2)]#
#= 1/[(PbarV^2 - a)/(RbarV^2)]#
#= color(green)((RbarV^2)/[PbarV^2 - a])#
Showing that these derivatives explicitly reduce to -1
Now we combine these to see if we get
#((delbarV)/(delT))_P((delT)/(delP))_(barV)((delP)/(delbarV))_T stackrel(?)(=) -1#
#= (RbarV^2)/[PbarV^2 - a] (barV)/R(-(RT)/(barV^2) + (2a)/(barV^3))#
#= (RbarV^2)/[PbarV^2 - a] (-(T)/(barV) + (2a)/(RbarV^2))#
#= -(TRbarV^2)/(barV(PbarV^2 - a)) + (2aRbarV^2)/(RbarV^2(PbarV^2 - a))#
#= -(TR^2barV^3)/(RbarV^2(PbarV^2 - a)) + (2aRbarV^2)/(RbarV^2(PbarV^2 - a))#
Now make sure that you recall that
#-x/(a + b) + y/(a + b) = (-(x - y))/(a + b) = -(x - y)/(a + b)# .
Thus:
#=> (-(TR^2barV^3 - 2aRbarV^2))/(RbarV^2(PbarV^2 - a))#
#= (-cancel(RbarV^2)(TRbarV - 2a))/(cancel(RbarV^2)(PbarV^2 - a))#
#= (-(RTbarV - 2a))/(PbarV^2 - a)#
Now we substitute
#=> (-((PbarV + a/(barV))barV - 2a))/(PbarV^2 - a)#
#= -(PbarV^2 + a - 2a)/(PbarV^2 - a)#
#= -cancel((PbarV^2 - a)/(PbarV^2 - a))#
#=> color(blue)(-1 = ((delbarV)/(delT))_P((delT)/(delP))_(barV)((delP)/(delbarV))_T)#
as expected!