Physical Chemistry: Adiabatic Expansion of Clouds at Altitude

In summary, the problem involves a cloud moving from an altitude of 2000m to 3500m and encountering a mountain. The cloud expands adiabatically with an initial temperature of 288K and a specific heat capacity of 28.86 J/Kmol. To find the final temperature, the ideal gas law and the relationship between pressure and volume for an adiabatic process are used. The final temperature is found to be 265K. The effect on moisture is not addressed in the conversation.
  • #1
whitebuffalo
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Homework Statement
Cloud moves from alt. 2000m (P = 0.802atm) to 3500m (P = 0.602) when it encounters a mountain. It expands adiabatically. The initial temp is 288K, CP,m for the air is 28.86 J/Kmol (assume ideal). What is the final temp and will it drop it's moisture?The attempt at a solution
This is what I did. CV,m = CP,m - R
CV,mdT = -[tex]\frac{RT}{V}[/tex]dV
[tex]\int[/tex]CV,mdT = [tex]\int[/tex]-[tex]\frac{RT}{V}[/tex]dV divide by T
CV,m[tex]\int[/tex][tex]\frac{1}{T}[/tex]dT = -R[tex]\int[/tex][tex]\frac{1}{V}[/tex]dV
CV,mln([tex]\frac{T2}{T1}[/tex]) = R ln([tex]\frac{V1}{V2}[/tex]) rearrange
T2 = T1(V1/V2)R/CV,m this to solve for T2, but 2 variables, so...

P1V1[tex]\gamma[/tex] = P2V2[tex]\gamma[/tex] solve for the ratio V1/V2
[tex]\frac{V1}{V2}[/tex] = ([tex]\frac{P2}{P1}[/tex])1/[tex]\gamma[/tex]

T2 = T1 (([tex]\frac{P2}{P1}[/tex])1/[tex]\gamma[/tex])R/CV,m

CV,m = CP,m - R = 20.546 [tex]\gamma[/tex] = CP,m/CV,m = 1.405

T2 = 288 (([tex]\frac{0.602}{0.802}[/tex])1/1.405)8.314/20.546 = 265KSomeone please tell me if I did this right, if not, what did I do wrong...and how do I know if it will drop moisture?

This is my first HW question in Pchem I and only the 2nd week of the semester, so please explain anything so I can understand it.
 
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  • #2
You should reverse the order of your first and second steps. Divide by T before integrating. Once the integral signs are there, you can't just divide to move variables from one side of the equation to the other. Your solution otherwise looks okay, but you could have done it more directly by simply using just the ideal gas law and the relationship [itex]PV^\gamma=c[/itex], where c is a constant.

I don't know about the moisture. Perhaps someone else can chime in on that part of the problem.
 

1. What is the adiabatic expansion of clouds at altitude?

The adiabatic expansion of clouds at altitude refers to the process by which clouds expand as they rise to higher altitudes in the atmosphere. This expansion is due to changes in air pressure and temperature, which can cause water vapor in the clouds to condense and form rain or snow.

2. How does adiabatic expansion affect cloud formation?

The adiabatic expansion of clouds is a key factor in the formation of precipitation. As the clouds rise to higher altitudes, the decrease in air pressure causes the air to cool and condense, resulting in the formation of rain or snow. This process is known as the adiabatic cooling effect.

3. What is the relationship between adiabatic expansion and altitude?

The relationship between adiabatic expansion and altitude is inverse. As altitude increases, air pressure decreases, and the clouds expand due to the decrease in pressure. The higher the altitude, the greater the adiabatic expansion and the more likely the formation of precipitation.

4. What factors can impact the adiabatic expansion of clouds at altitude?

Several factors can impact the adiabatic expansion of clouds at altitude, including the initial temperature and humidity of the air, the amount of water vapor present, and the altitude at which the clouds form. Additionally, atmospheric conditions such as wind and temperature inversions can also affect the adiabatic expansion of clouds.

5. How does adiabatic expansion impact weather patterns?

The adiabatic expansion of clouds at altitude plays a crucial role in weather patterns. As clouds expand and cool, precipitation is formed, which can lead to rain, snow, or other forms of precipitation. This can have a significant impact on local and regional weather patterns, as well as the overall climate of an area.

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