Calculating maximum amount of water vapor per unit volume

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SUMMARY

The discussion focuses on calculating the maximum amount of water vapor per unit volume in air at two different temperatures: 288 K at the surface and 220 K at 10 km altitude. The relevant equations include the saturation vapor pressure equation, e_{s}=Ae^{\beta T}, and the vapor density equation, e=\rho_{v}R_{v}T. The user initially struggles with unit consistency and arrives at an incorrect value of 1,103,248.397 kg/m³ instead of the expected 0.0126 kg/m³ for the surface temperature. The ideal gas law, pV=nRT, is suggested as a necessary tool to resolve the confusion regarding pressure and vapor density.

PREREQUISITES
  • Understanding of thermodynamics and the ideal gas law
  • Familiarity with saturation vapor pressure calculations
  • Knowledge of specific gas constants, particularly for water vapor (R_{v})
  • Basic algebra for manipulating equations and unit conversions
NEXT STEPS
  • Study the derivation and application of the ideal gas law in atmospheric science
  • Learn about the calculation of saturation vapor pressure using the Clausius-Clapeyron equation
  • Explore the relationship between temperature, pressure, and density in gas mixtures
  • Investigate the concept of specific humidity and its relevance in meteorology
USEFUL FOR

Students in atmospheric science, meteorology, or environmental science, as well as anyone involved in calculations related to humidity and vapor pressure in air.

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Homework Statement



Compute the maximum amount of water vapor per unit volume that air can hold at the surface, where Ts = 288 K, and at a height of 10 km where T = 220 K. Express your answers in kg m-3.

Homework Equations



e_{s}=Ae^{\beta T}

e=\rho _{v}R_{v}T

The Attempt at a Solution



Since saturation occurs when e=e_{s}, I figured I would set the two equations equal to each other. However, solving for \rho _{v} doesn't work... The units don't work out, and I get a really large number... I feel like I have to somehow relate this to the total pressure of the air, but I'm unsure how to go about this.
 
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I was incorrect in my previous post saying that the units didn't work out... Solving for {\rho _{v}} when e=e_{s} does produce an answer in \frac{kg}{m^{3}}... However I'm getting 1,103,248.397 \frac{kg}{m^{3}}, for the first case where T=288 K, which is way off from what I should be getting (0.0126 \frac{kg}{m^{3}}).

I believe I then have to use the ideal gas equation, pV=nRT, plugging in p for e... But this is where the confusion comes in. Hopefully someone can help me with this tonight, since this HW is due tomorrow morning...
 

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