Adiabatic Process: Derive Temp Change w/ Pressure

In summary: Using this, I can then express the ideal gas law PV=nRT in terms of T and P, and then substitute this into the adiabatic equation PV^{k}=constant to solve for the change in temperature with pressure. Finally, I can use this derived equation to calculate the final temperature of the air parcel after being lifted adiabatically from 1000hPa to 800hPa, starting at an initial temperature of 20^{o}C.
  • #1
KingBigness
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Homework Statement



Derive an expression for the change in temperature with pressure for an adiabatic process. If a dry air parcel is lifted adiabatically from 1000hPa to 800hPa what is the final temperature if it was initally at 20[itex]^{o}[/itex]C?[itex]^{}[/itex]

Homework Equations


[itex]PV^{k}=constant[/itex]
I derived an equation [itex]\frac{T_{2}}{T_{1}}= \frac{P_{2}}{P_{1}}^{\frac{k-1}{k}}[/itex]

Not sure if this is the correct equation?

The Attempt at a Solution



[itex]k = \frac{c_{p}}{c_{v}}[/itex]

I'm not sure if this is even the right direction for this question, and if it is where can I get the value of the c's?

Any help would be appreciated. thanks =D
 
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  • #2
KingBigness said:

Homework Statement



Derive an expression for the change in temperature with pressure for an adiabatic process. If a dry air parcel is lifted adiabatically from 1000hPa to 800hPa what is the final temperature if it was initally at 20[itex]^{o}[/itex]C?[itex]^{}[/itex]

Homework Equations


[itex]PV^{k}=constant[/itex]
I derived an equation [itex]\frac{T_{2}}{T_{1}}= \frac{P_{2}}{P_{1}}^{\frac{k-1}{k}}[/itex]

Not sure if this is the correct equation?
What is your reasoning? Assume air is an ideal gas.

Express V in terms of P and T and substitute into [itex]PV^{k}=constant[/itex]

AM
 

1. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no heat transfer between the system and its surroundings. This means that the system's internal energy remains constant.

2. How is temperature change related to pressure in an adiabatic process?

The temperature change in an adiabatic process is directly proportional to the change in pressure. This can be derived using the ideal gas law, where pressure and temperature are inversely related.

3. What is the formula for calculating temperature change in an adiabatic process?

The formula for calculating temperature change in an adiabatic process is ΔT = -(γ-1) * T * ΔP / P, where ΔT is the change in temperature, γ is the heat capacity ratio, T is the initial temperature, ΔP is the change in pressure, and P is the initial pressure.

4. How does the heat capacity ratio affect temperature change in an adiabatic process?

The heat capacity ratio, represented by the symbol γ, is a constant value that depends on the type of gas being used. It affects temperature change in an adiabatic process by determining how much the temperature will change in response to a change in pressure. A higher heat capacity ratio means a greater change in temperature for a given change in pressure.

5. Can the temperature change be negative in an adiabatic process?

Yes, the temperature change can be negative in an adiabatic process. This occurs when there is a decrease in pressure, which results in a decrease in temperature. Similarly, an increase in pressure will result in an increase in temperature. The magnitude of the temperature change is dependent on the heat capacity ratio and the initial conditions of the system.

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