The Adiabatic Equation 531
20.7 The Adiabatic Equation
If an amount of gas at high pressure and temperature is placed in a container fitted with a piston, the gas will push the piston outward and do work on it. Such a process of expansion converts thermal energy into useful mechanical energy -- the temperature of the gas decreases as it delivers work to the piston. This process is at the core of the operation of steam engines, automobile engines, and other heat engines.
In this section we will investigate the equations for the expansion of a gas. We will assume that the gas is thermally insulated, so it neither receives heat from its environment nor loses any. The temperature decrease of the gas is then entirely due to the work that the gas does on its environment. Such a process occurring without exchange of heat with the environment is called adiabatic.
If the volume of the gas increases by a small amount dV, the work done by the gas on this piston is [see Eq. (10)]
The heat absorbed is zero; hence the change of energy of the gas is
The change of energy can also be expressed in terms of the change of temperature [see Eq. (8)]:
Combining Eqs. (23) and (24), we have
By differentiating the ideal-gas law nRT = pV, we find
If we eliminate dT between Eqs. (25) and (26), we obtain
Let us write this as
The quantity g is the ratio of the two kinds of heat capacities; for instance, for an ideal monoatomic gas, g = Cp/Cv = 5/3. To obtain an equation linking p and V, we must integrate Eq. (29). The integrals of each side are natural logarithms with an additive constant of integration:
Using the properties of logarithms, we can rewrite this as
This is the equation for the adiabatic expansion of a gas. From this we can readily calculate the drop of temperature that the gas suffers as it expands. According to the ideal-gas law, pV µ T, and thus Eq. (33) becomes
which shows that the temperature is inversely proportional to Vg-1. Figure 20.16 shows the adiabatic curves pVg = [constant] in a p-V diagram for an ideal monoatomic gas, with g = 5/3. For comparison the figure also shows the isothermal curves T = [constant], or pV = [constant]. Note that the adiabatic curves are steeper than the isothermal curves. When a gas expands adiabatically, it evolves downward along one of the adiabatic curves, and it crosses into regions of lower temperature.
The decrease of temperature of an expanding gas can be perceived quite readily when air is allowed to rush out of the valve of an automobile tire; this expanding air feels quite cool. The expansion process is approximately adiabatic because the rushing air, although not insulated from its surroundings, expands so quickly that it does not have time to exchange heat with the surrounding atmospheric air. Conversely, the increase of temperature during the adiabatic compression of air can be perceived when operating a manual air pump. The compression of air in the barrel of the pump produces a noticeable warming of the pump.
EXAMPLE 5. In one of the cylinders of an automobile engine, after the explosive combustion of the fuel, the gas has an initial pressure of 3.4 X 106 N/m2 and an initial volume of 50 cm3. As the piston moves outward, the gas expands nearly adiabatically to a final volume of 250 cm3. What is the final pressure? Assume that g = 1.40.
SOLUTION: By Eq. (33),
EXAMPLE 6. How much work does the gas described in the preceding example do during its adiabatic expansion?
SOLUTION: The work done during a small increase of volume is p dV and the work done during the entire expansion is
According to Eq. (33),
With the numbers of Example 5, this gives