Relation between change of pressure and temperature in adiabatic process

In summary, the relationship between pressure and temperature in an adiabatic process is described by the adiabatic equation, which states that for an ideal gas, the product of pressure and volume raised to the adiabatic index (γ) is constant. In this process, when a gas expands without heat exchange, its temperature decreases, and when it is compressed, its temperature increases. The equation \( PV^\gamma = \text{constant} \) and the relation \( T V^{\gamma-1} = \text{constant} \) illustrate how changes in pressure and volume affect temperature, emphasizing that in an adiabatic process, temperature changes are directly linked to pressure changes.
  • #1
T C
340
8
TL;DR Summary
I am recently trying to formulate the relation between change in pressure and temperature for adiabatic process. Just submitting the math below for understanding of others. If there is any fault, then kindly rectify me.
In case of adiabatic process, we all know that the relation between temperature and pressure and that's given below:​
P. T(γ/(1-γ)) = Const.
therefore, P = Const. T(γ/(γ - 1))
or, ΔP = Const. (γ/(γ - 1)).ΔT(1/(γ - 1))​
It's just an attempt to find out the relation. Don't know how much correct I am. Waiting for comments from others.​
 
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  • #2
No, it is:
$$P_0T_0^k = C$$
and:
$$PT^k = C$$
Therefore
$$P_0T_0^k = PT^k$$
Where ##k = \frac{\lambda}{1-\lambda}##. Thus:
$$\Delta P = P- P_0$$
$$\Delta P= P_0\left(\left(\frac{T_0}{T}\right)^k-1\right)$$
$$\Delta P = P_0\left(\left(\frac{T}{T_0}\right)^{-k}-1\right)$$
$$\frac{\Delta P}{P_0} = \left(\frac{T_0 + \Delta T}{T_0}\right)^{-k}-1$$
Or:
$$1+ \frac{\Delta P}{P_0} = \left(1+\frac{\Delta T}{T_0}\right)^{\frac{\lambda}{\lambda - 1}}$$
 

FAQ: Relation between change of pressure and temperature in adiabatic process

1. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which no heat is exchanged between a system and its surroundings. This means that any change in the internal energy of the system is due solely to work done on or by the system, rather than heat transfer.

2. How does pressure change with temperature in an adiabatic process?

In an adiabatic process, the relationship between pressure and temperature can be described by the adiabatic equations. For an ideal gas, the pressure and temperature are related through the equation \( P \propto T^{\frac{\gamma}{\gamma - 1}} \), where \( \gamma \) is the heat capacity ratio (Cp/Cv). This indicates that as the temperature increases, the pressure also increases, provided the volume remains constant.

3. What is the significance of the adiabatic index (gamma)?

The adiabatic index, denoted as \( \gamma \), is the ratio of the heat capacity at constant pressure (Cp) to the heat capacity at constant volume (Cv). It plays a crucial role in determining the relationship between pressure, volume, and temperature during an adiabatic process. Different gases have different values of \( \gamma \), which affects how they respond to changes in pressure and temperature.

4. What happens to temperature and pressure during adiabatic expansion?

During adiabatic expansion, a gas does work on its surroundings as it expands, leading to a decrease in internal energy. As a result, both the temperature and pressure of the gas decrease. This is because the gas does not absorb heat from the surroundings to compensate for the work done, resulting in cooling.

5. How is the adiabatic process different from an isothermal process?

The main difference between an adiabatic process and an isothermal process is that in an adiabatic process, there is no heat exchange with the surroundings, while in an isothermal process, the temperature remains constant through heat exchange. In an isothermal process, any work done by or on the system is balanced by heat transfer, keeping the internal energy constant, whereas in an adiabatic process, changes in internal energy occur solely due to work done.

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