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consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant

let $n=R=1$

differentiate with respect to $t$ (time)

$dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * dV/dt......................∂P/∂T = 1/V , ∂P/∂V = -T/V2$

considering an experiment where i can adjust the rate $dV/dt$ and $dT/dt$ as i want

  1. find $V(t)$ and $T(t)$ that give maximum $dP/dt$ at point $V=10 cm^3 , T = 25$ degree
  2. find $V(t)$ and $T(t)$ that give maximum $dP/dt$ at point $V=V_0 cm^3 , T = T_0$ degree

from Ideal gas law

PV=nRT

where P = pressure , V= volume , T = temperature, n= constant=1 , R = constant =1 let n = R = 1 , we have P=T/V

what rate should the temperature and volume be changing to make the rate of change of pressure fastest at point T = 25 , V = 10 ?

differentiate with respect to t

dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * dV/dt......................∂P/∂T = 1/V , ∂P/∂V = -T/V2

if i set T = 25 +t , V = 10+2t , i.e tempertature starting at 25 degree Celcius with increment of 1 degree per minute and volume of 10 cm3 with increment of 2 cm3 per minute

from this point ∇f(x) would be < 1/V , -T/V2> = < 1/10 ,-25/100>

and U→ = < dT/dt, dV/dt > = < 1,2 >

Ok now, if i substitute all above and calculate dP/dt = D(1,2)(25,10) = 1/10-1/2 = -4/5 so the pressure is decreasing at rate of -4/5 unit when volume is raised 2 times faster than temperature at the point V= 10 cm3 , T= 25 degree

∇f(x)= < 1/10 ,-25/100> is the direction where rate of change of pressure is fastest which is

√(1/100+1/16) = 0.27 (i dont even know if it's decreasing or increasing at this rate) and it's less than 4/5 if i scale down <1 , 2> to unit vector it will be -0.18 but V(t) = 10+2t/√5 same apply to T(t) also

you see my point?

here are list of questions

  1. Is pressure increasing or decreasing at the direction of ∇f(x)
  2. How fast should i change my temperature and volume(suppose im doing an experiment where i can change the rate of these 2 factors) to give the maximum dP/dt at the given initial point T=T0 and V=V0
  3. is there always a minimum dP/dt in the direction of -∇f(x)?

4.what does scaling the magnitude of ∇f(x) and U up and down mean(which of course, affect the value of dP/dt) to my experiment does it make any sense in physical point of view?

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  • $\begingroup$ please help i have been thinkink about this problem for more a week $\endgroup$ – kochibacha Jul 5 '14 at 11:39
  • $\begingroup$ Can't you choose any value of $\dot{V} = \frac{dV}{dt}$ and $\dot{T} = \frac{dT}{dt}$ you want? Wouldn't that mean that $\frac{dP}{dt} = -\frac{T}{V^2}\dot{V} + \frac{1}{V}\dot{T}$ In which case an extremely fast increasing temperature combined with an extremely fast decreasing volume would create the greatest positive pressure change--regardless of initial conditions (which makes since if you think about it). $\endgroup$ – Jared Jul 8 '14 at 8:44
  • $\begingroup$ first please recall this theorem (tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx) The maximum value of Df(x) (and hence then the maximum rate of change of the function f(x) ) is given by ||∇f(x)|| and will occur in the direction given by ∇f(x). so this theorem cannot be applied to real physical problem? because I can make V' and T' to any larger number? $\endgroup$ – kochibacha Jul 8 '14 at 9:20

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