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…
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.
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Q1: How is work done calculated in an adiabatic process?
In an adiabatic process, work done can be calculated using the internal energy change expressed in terms of temperature change. Alternatively, use the adiabatic condition relating initial and final pressures and volumes. By integrating the work expression during volume change with the pressure relationship, you can determine the total work done by the gas during adiabatic expansion or compression.
Q2: What happens to pressure and temperature when a gas undergoes adiabatic compression?
During adiabatic compression, both pressure and temperature of the gas increase significantly. As the volume decreases, the gas is compressed without heat exchange, causing internal energy to rise and temperature to increase. This temperature increase results in higher pressure, which is why diesel engines achieve fuel ignition without spark plugs through adiabatic compression of the air-fuel mixture.
Q3: Why does a weather balloon expand as it rises in the atmosphere?
A weather balloon expands adiabatically as it rises because atmospheric pressure decreases with altitude. With no heat exchange occurring, the gas inside expands to fill a larger volume. The decreasing external pressure allows the helium gas to expand from its initial volume to a final larger volume while the pressure and temperature of the gas decrease.
Q4: How does adiabatic compression differ from isothermal compression?
In adiabatic compression, no heat is exchanged with surroundings, so temperature and pressure increase significantly as volume decreases. In isothermal compression, temperature remains constant while heat is removed, resulting in lower final pressure. Adiabatic compression produces higher pressure and temperature changes, which is why it enables spontaneous fuel ignition in diesel engines.
Q5: What does the negative sign indicate in adiabatic work calculations?
A negative work value indicates that work is done on the gas rather than by the gas. During adiabatic compression, the piston performs work on the gas mixture, compressing it to a smaller volume. The negative sign distinguishes compression work from expansion work, where positive values represent work done by the expanding gas on its surroundings.
Q6: How can you find final pressure and temperature in an adiabatic process if initial conditions are known?
Use the adiabatic condition equation relating pressure and volume with the molar heat capacity ratio (gamma) of the gas. Substitute known initial pressure, volume, and final volume to calculate final pressure. Then apply the ideal gas law using the calculated final pressure to determine final temperature. This approach solves adiabatic compression problems in engines and atmospheric systems.
Q7: What role does the heat capacity ratio play in adiabatic processes?
The molar heat capacity ratio (gamma), typically 1.4 for air, is essential in adiabatic process equations relating pressure, volume, and temperature. This dimensionless ratio appears in the adiabatic condition and determines how significantly pressure and temperature change during compression or expansion. Different gases have different gamma values, affecting their adiabatic behavior.