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 expans…
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.
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Q1: What happens to temperature when air expands in an adiabatic process?
During adiabatic expansion, temperature drops as air molecules expand and do work on their surroundings without receiving heat. This temperature decrease is described by the temperature-volume relationship for adiabatic processes. When rising air expands due to decreasing atmospheric pressure, this cooling causes water vapor to condense, forming clouds in the atmosphere.
Q2: How is the pressure-volume relationship derived for an adiabatic process?
The pressure-volume relationship is derived by combining the first law of thermodynamics with the ideal gas law. Starting with internal energy change expressed through molar heat capacity, differentiating the ideal gas law, and equating temperature change expressions yields an equation involving molar heat capacities. After simplification using the ratio of heat capacities and integration, the final condition pVγ = constant emerges for adiabatic processes.
Q3: What is the key difference between free expansion and adiabatic expansion?
In free expansion, no work is done and internal energy remains unchanged, so initial and final temperatures are equal. In adiabatic expansion, work is performed and internal energy changes, causing temperature to decrease. Additionally, the pressure-volume relationship pVγ = constant holds for adiabatic processes but not for free expansion, even though both are adiabatic.
Q4: Why can only initial and final states be plotted for free expansion on a pV diagram?
During free expansion, the gas is in equilibrium only at the initial and final points. The expansion process itself occurs through non-equilibrium states that cannot be plotted on a pV diagram. In contrast, adiabatic expansion maintains equilibrium throughout, allowing the entire path to be represented on the diagram as a curve following the pVγ = constant condition.
Q5: How do pressure and volume change in free expansion of an ideal gas?
In free expansion of an ideal gas, temperature remains constant since internal energy does not change. Because temperature is constant for an ideal gas, the product of pressure and volume (pV) also remains constant. This means initial and final states lie on the same isotherm, with pressure and volume adjusting proportionally to maintain constant pV.
Q6: What mathematical expressions describe adiabatic processes for ideal gases?
Three equivalent mathematical expressions describe adiabatic processes: pVγ = constant (pressure-volume), TVγ-1 = constant (temperature-volume), and pT-γ/(γ-1) = constant (pressure-temperature). These conditions are derived from the first law of thermodynamics combined with the ideal gas law. Each expression relates different pairs of state variables while maintaining the adiabatic constraint.
Q7: How does the molar heat capacity ratio simplify the adiabatic process equation?
The molar heat capacity ratio, gamma (γ), is defined as the ratio of heat capacity at constant pressure to heat capacity at constant volume. Using this ratio simplifies the complex equation derived from combining thermodynamic principles into the elegant form pVγ = constant. This simplification makes it easier to analyze and predict gas behavior during adiabatic processes without tracking individual heat capacity values.