1. The problem statement, all variables and given/known data Combine dP/dz=-?gp/

Question

1. The problem statement, all variables and given/known data
Combine dP/dz=-?gp/RT and the vapor pressure to find the rate dT/dz for the change in the boiling temperature of water at varying altitude above sea level.  After solving it algebraically, assuming that the latent heat of vaporization of water L=2.4x106 J/kg and the density of water vapor ?~0.6 kg/m3, find the rate of change of T in Kelvin per kilometer. (Hint: apply the chain rule)


2. Relevant equations
dP/dz=-?gp/RT, ? is the molecular weight
dT/dz


3. The attempt at a solution
My textbook says the definition of vapor pressure is p=p0e(-L/RT).  However, to get there they used the Clausius-Clapeyron equation dP/dT=L/T?V and one of the intermediate steps is 1/p(dp/dT)=L/RT2.   This is the equation I used.  Applying the chain rule, dT/dz=(dp/dz)(dT/dp) I found dT/dz=-?gT/L.  However, when asked to find an actual value of dT/dz, I am given the L and ? (density) to plug into the equation.  Does ? have something to with the density?  And what would I use for T.  I think I may have done something wrong.  I tried working out the equation in a different way using the original Clausius-Clapeyron equation for the vapor pressure: dP/dT=L/T?V and found dT/dz=-?gp?V/RL, but am not sure how to use density, ?, and L, latent heat, with this equation either.  Am I using the right equation for vapor pressure?  Missing a step?  How can I account for ? and L?  Thanks for any of your help!

Explanation / Answer

you are right. you have to just extend it

dT/dz=-?gT/L

so int(dT/T) = int (-?g/L)dh {intgration from h = 0 to h and T=To to T, where To is boiling temp. at h=0 mean 100oC}

so ln(T/To) = -?gh/L

T = Toe^(-?gh/L)

and in dT/dz=-?gT/L

T = To

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.