20.16:
Work Done in an Adiabatic Process
Consider a weather balloon filled with helium gas having some initial pressure and volume rising and expanding adiabatically to a final volume under decreasing atmospheric pressure. What is the work done by the gas?
If the initial and final temperatures are known, then, for an adiabatic process, the work done is obtained from its internal energy, expressed in terms of temperature change.
Another way is using the condition for the adiabatic process, from which the gas's initial and final pressures and volumes can be related. The terms are rearranged for simplification.
Recall the work during the volume change and substitute the obtained pressure expression in it.
By integrating this expression with the volume change, the work done during an adiabatic process can be determined.
If the initial gas pressure in the helium balloon is 1 atm, and the balloon's volume changes from 25 m3 to 100 m3, then by substituting the known quantities in the derived expression, the work done by the gas can be calculated.
20.16:
Work Done in an Adiabatic Process
Consider the adiabatic compression of an ideal gas in the cylinder of an automobile diesel engine. The gasoline vapor is injected into the cylinder of an automobile engine when the piston is in its expanded position. The temperature, pressure, and volume of the resulting gas-air mixture are 20 °C, 1.00 x 105 N/m2, and 240 cm3 , respectively. The mixture is then compressed adiabatically to a volume of 40 cm3. Note that, in the actual operation of an automobile engine, the compression is not quasi-static, although we are making that assumption here. What are the pressure and temperature of the mixture after the compression? What is the work done during the compression?
Solution:
For an adiabatic compression process the expression for pressure is written as
Using the molar heat capacity ratio of air as 1.4 and by substituting the known quantities in the expression (1), the pressure of the mixture after the compression is calculated.
From the ideal gas law and using the obtained pressure value, the temperature of the mixture after the compression is also calculated.
Thus, the pressure and temperature of the mixture after compression inside the cylinder of an automobile diesel engine are determined.
If the compression process is isothermal, then the value of pressure changes from the obtained value. This is because, in an adiabatic compression process, if the temperature increases, the pressure becomes greater. Hence, on injecting the fuel into the cylinder near the compression stroke, the air mixture attains a higher temperature, resulting in spontaneous ignition of the fuel without any need for spark plugs.
By knowing the initial and final pressure, the work done during the compression can be obtained using
Substituting the known and the obtained quantities in the above expression, the work done equals -1.63 J. Here, the negative sign indicates that the work is done by the piston on the air-gas mixture.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Pp. 641, 642
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-2/pages/3-6-adiabatic-processes-for-an-ideal-gas