20.15:
Pressure and Volume in an Adiabatic Process
Consider a volume of air molecules rising in the atmosphere. As the pressure decreases with altitude, the air molecules try to maintain the same pressure by increasing their volume under an adiabatic process.
The internal energy change is written in terms of molar heat capacity and rearranged.
Using the ideal gas law, differentiating it, and rearranging the terms gives an expression for temperature change.
Equating the temperature change equations obtained from the first law of thermodynamics and the ideal gas law and rearranging gives an equation in terms of the molar heat capacity.
Divide throughout by pV and use the relation between molar heat capacities. The equation can be further simplified using their ratio.
Upon integration and using a general mathematical expression for logarithms, the condition for an ideal gas to undergo an adiabatic process is obtained in terms of pressure and volume.
Similarly, the conditions for adiabatic processes in terms of temperature-volume and pressure-temperature can be written.
So, when air expands, the temperature drops as given by the temperature-volume condition, condensing to form clouds.
20.15:
Pressure and Volume in an Adiabatic Process
Free expansion of a gas is an adiabatic process. However, there are few differences between free expansion and adiabatic expansion. During free expansion, no work is done, and there is no change in internal energy. But, for an adiabatic expansion, work is done, and there is a change in internal energy. During an adiabatic process, the relation between the pressure and volume is obtained from the condition for the adiabatic process, that is,
However, for a free expansion process, this condition does not hold even if the expansion is adiabatic. Also, in a free expansion process, the gas is in an equilibrium state only at the initial and final points. Thus, on a pV diagram, only those initial and final points can be plotted, not the expansion itself. In addition to this, the initial and final temperature for an adiabatic process are related by the expression,
But for the free expansion, there is no change in internal energy; hence, the initial and final temperatures are equal. Thus, instead of the above equation, we have
From the expression for temperature, the initial and final states of a free expansion lie on the same isotherm in a pV diagram. Now, if the gas is assumed to be ideal, then, as there is no change in the temperature, there can be no change in the product of pressure and volume. Hence for the free expansion, the pressure and volume can be expressed as,
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. P. 640, 641
- Walker, Jearl (2007). Halliday and Resnick Fundamentals of Physics. Wiley. Pp 573-574