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
- find $V(t)$ and $T(t)$ that give maximum $dP/dt$ at point $V=10 cm^3 , T = 25$ degree
- 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
- Is pressure increasing or decreasing at the direction of ∇f(x)
- 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
- 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?