Calculating Changes in Internal Energy for a Rising Air Parcel

[il27]

Homework Statement

A 1 kg parcel of dry air initially at a temperature of 15 Celsius (288 K) rises from ground level to its equilibrium height, which corresponds to a pressure of 750 hPa.

1. What is the decrease in internal energy?
2. What is the temperature of the air parcel at its equilibrium height?
3. Assuming the parcel ascends at the dry adiabatic lapse rate, what is the height of the parcel at
   

equilibrium?

Homework Equations

pV = mRT
dU/dT = cV

The Attempt at a Solution

[/B]
pV = mRT

R (dry air gas constant) = 287.056 J/kg*K

V = (mRT)/p

W = p∆V
W = -∆U

I am not sure if you are able to just plug the pressure (750 hpa), mass, temperature (288K), into the volume equation like that...

 

[Chestermiller]

In hPa, what is the pressure at ground level?

Since this is an adiabatic reversible expansion of the parcel of air, from the first law, how are the differential changes in internal energy related to the differential changes in parcel volume? Treating the air as an idea gas, how are the differential changes in internal energy related to the differential changes in parcel temperature?

 

[il27]

In hPa, what is the pressure at ground level?

Since this is an adiabatic reversible expansion of the parcel of air, from the first law, how are the differential changes in internal energy related to the differential changes in parcel volume? Treating the air as an idea gas, how are the differential changes in internal energy related to the differential changes in parcel temperature?

hPa at ground level will be 1013.25? which equation would be the right one to start with?
Also can i find the volume at ground level using V = (mRT)/p?

 

[Chestermiller]

hPa at ground level will be 1013.25? which equation would be the right one to start with?
Also can i find the volume at ground level using V = (mRT)/p?

I can help you with this, but, before we begin, I need to know if you have any kind of experience solving problems involving adiabatic reversible expansion of ideal gases. Please fill me in on what you know.

 

[il27]

I can help you with this, but, before we begin, I need to know if you have any kind of experience solving problems involving adiabatic reversible expansion of ideal gases. Please fill me in on what you know.

Yes, I have some experience. I have dealt with the Clausius Clapeyron equation and used work and energy with regards to thermodynamics. One question, can I assume the pressure at ground level is 1013.25 hpa and then use the Poisson equation? Of (T/T0) = (p/p0)^k? and then isolate for T to find the temperature at the equilibrium point?

 

[Chestermiller]

Yes, I have some experience. I have dealt with the Clausius Clapeyron equation and used work and energy with regards to thermodynamics. One question, can I assume the pressure at ground level is 1013.25 hpa and then use the Poisson equation? Of (T/T0) = (p/p0)^k? and then isolate for T to find the temperature at the equilibrium point?

Yes. This is the pressure to use. Also, the equation you have written is the correct one to use to get the temperature, provided $$k=\frac{\gamma-1}{\gamma}$$ where $\gamma$ is the ratio of the specific heat at constant pressure to the specific heat at constant volume.

Very good so far.

 

[il27]

Yes. This is the pressure to use. Also, the equation you have written is the correct one to use to get the temperature, provided $$k=\frac{\gamma-1}{\gamma}$$ where $\gamma$ is the ratio of the specific heat at constant pressure to the specific heat at constant volume.

Very good so far.

Excellent! Thank you so much!
I was then thinking of using this equation to find the height that the air parcel is at when reaching equilibrium:
$$ T = T_0 - \Gamma*Z $$ and then isolating Z to find the new height. Is that in the right direction to finding Z?

 

[Chestermiller]

Excellent! Thank you so much!
I was then thinking of using this equation to find the height that the air parcel is at when reaching equilibrium:
$$ T = T_0 - \Gamma*Z $$ and then isolating Z to find the new height. Is that in the right direction to finding Z?

Yes, for part c, that's what I would do.

 

[il27]

Yes, for part c, that's what I would do.

Thank you! I'm so sorry, but just one last question about part A. What would be the best method to find the decrease in internal energy? I am a bit confused on that part.
Thank you for all your help :)

 

[Chestermiller]

$$\Delta U=mC_v\Delta T$$