DOC PREVIEW
UCSD SIO 217A - Lecture

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

1!Lecture Ch. 4b!• Hydrostatic equilibrium!– Special cases!– Pressure altitude dependence!• More Midterm Review problems!– Terminology review!Curry and Webster, Ch. 4 (pp. 96-115; skip 4.5 (except 4.5.1), 4.6)!Tuesday, Oct. 20: Homework, Review (bring questions), and Read Ch. 5!Tuesday, Oct. 27: Midterm!Thursday, Oct. 29: meet to work on ROAST!!Water Vapor Metrics!Special Cases of Hydrostatic Equilibrium!• 1. rho=constant (homogeneous)!– H=8 km =RT/g=scale height eq. 1.39!• 2. constant lapse rate (implied if hydrostatic, homogeneous, and ideal gas)!– -dT/dz=constant=-g/R=-34/deg/km!• 3. isothermal T=constant (and ideal gas)!– p=p_0*exp(-z/H)!Special Cases of Hydrostatic Equilibrium!• Hydrostatic: Force balance on gravity and upward pressure!Homogeneous Atmosphere!Homogeneous Constant density Constant lapse rate Isothermal Atmosphere!2!Hydrostatic Equilibrium Example!Consider a planet with an atmosphere in hydrostatic equilibrium. Assume that theatmosphere is an ideal gas. Also assume that the temperature is a maximum at thesurface of the planet, and, as height increases, the temperature in the atmospheredecreases linearly (in other words, temperature decreases with height at a constant rate).Derive a formula for atmospheric density as a function of height in this atmosphere.Curry and Webster, Ch. 4!Quiz !• If a homogeneous mixture of two substances coexists in liquid and vapor phases at a series of pressures (each of which corresponds to exactly one temperature), how many degrees of freedom are in this system?!• What is thermal equilibrium? Give an equation.!• Give the equation for the Gibbs phase rule. !• What is the change in free energy for a phase change?!• What type of pressure change is described by the Clausius-Clapeyron equation? i.e. what changes as a function of what under what conditions?!Answer briefly and clearly, with appropriate equations or diagrams. !Curry and Webster, Ch. 4!Answers!• If a homogeneous mixture of two substances coexists in liquid and vapor phases at a series of pressures (each of which corresponds to exactly one temperature), how many degrees of freedom are in this system? !– 1!• What is thermal equilibrium? Give an equation. !– T1=T2!• Give the equation for the Gibbs phase rule. !– f=c+2-p!• What is the change in free energy for a phase change? !– 0 (at constant T and P)!• What type of pressure change is described by the Clausius-Clapeyron equation? i.e. what changes as a function of what under what conditions? !– ps=f(T) for l/v saturation pressure!Hydrostatic Equilibrium Example!Consider a planet with an atmosphere in hydrostatic equilibrium. Assume that theatmosphere is an ideal gas. Also assume that the temperature is a maximum at thesurface of the planet, and, as height increases, the temperature in the atmospheredecreases linearly (in other words, temperature decreases with height at a constant rate).Derive a formula for atmospheric density as a function of height in this atmosphere.From the hydrostatic equation for an ideal gas (Eqn. 1.42)€ ∂p = −pgRdT∂zand a constant lapse rate € Γ = −dTdz we get€ dp = −pgRdTdz−dTdzΓ        = −pgΓRddTTdpp=gΓRd      dTTdpp=gΓRd      dTT∫∫lnpp0=gΓRd      lnTT0p = p0TT0      gRdΓ      which is Eqn 1.48. Then dividing both sides by RT and noting that for an ideal gas€ ρ=pRT, we get€ pRT=ρ=p0RTTT0      gRdΓ=p0R T0− Γz( )T0− ΓzT0      gRdΓClausius Clapeyron Example!The saturation vapor pressure at a temperature of 30°C is 42.4 hPa. The gas constant fordry air is 287 J K-1 kg-1. The gas constant for water vapor is 461 J K-1 kg-1.In addition to the constants given above, here is one more: the saturation vapor pressureat a temperature of 40°C is 73.8 hPa. Assuming that the latent heat of vaporization isconstant, use this information to calculate the numerical value for this latent heat.The Clausius Clapeyron equation can be integrated if L is assumed constant, and theresult is Eqn. 4.23. Using 30°C=303K and 40°C=313K, and knowing saturationvapor pressure values for each, the only unknown is L. Solving Eqn. 4.23,€ e2= e1exp −LlvRv1T2−1T1            Llv= −RvT1T2T1− T2      lne2e1      = −461303 × 313303 − 313      ln73.842.4      = 2.4 ×106L=2.4x106 J/kg.Degrees of Freedom Example!Name the five main components of the atmosphere. (a) If all components are in the gasphase, how many degrees of freedom are there in the system? (b) If water condenses orfreezes, does that number increase or decrease? (c) If new components are added bypollution, how does that change (i) the number of possible phases and (ii) the degrees offreedom of the atmosphere?The five main components of the atmosphere are nitrogen (N2), oxygen (O2), carbondioxide (CO2), argon (Ar), and water (H2O).(a) For this system, we can use the Gibbs phase rule (Eqn. 4.2) with χ=5, ϕ=1, f=χ-ϕ+2=6.(b) Condenses ϕ=2, f=5 [decrease]; freezes ϕ=2, f=5 [decrease] (both: ϕ=3, f=4[decrease]).(c) (i) number of phases that can exist at atmospheric pressure may increase withadditional components, since multiple liquid and solid phases may form; (ii)degrees of freedom increase with the number of components and will decreasewith the number of phases.3!Curry and Webster, Ch. 4!Quiz !• If a pure substance coexists in liquid and vapor phases at a series of pressures (each of which corresponds to exactly one temperature), how many degrees of freedom are in this system?!• What is the relationship between two pressures at mechanical equilibrium?!• Give the equation for the Gibbs phase rule. !• What three types of equilibrium are required by the Gibbs phase rule? !Answer briefly and clearly, with appropriate equations or diagrams.


View Full Document

UCSD SIO 217A - Lecture

Download Lecture
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?