19.7.3 Adiabatic Processes

Section 19.4 introduced the idea of an adiabatic process, a process in which no heat energy is transferred $(Q = 0)$ . FIGURE 19.19 compares an adiabatic process with isothermal and isochoric processes. We’re now prepared to look at adiabatic processes in more detail.

FIGURE 19.19 Full Alternative Text

In practice, there are two ways that an adiabatic process can come about. First, a gas cylinder can be completely surrounded by thermal insulation, such as thick pieces of Styrofoam. The environment can interact mechanically with the gas by pushing or pulling on the insulated piston, but there is no thermal interaction.

Second, the gas can be expanded or compressed very rapidly in what we call an adiabatic expansion or an adiabatic compression. In a rapid process there is essentially no time for heat to be transferred between the gas and the environment. We’ve already alluded to the idea that heat is transferred via atomic-level collisions. These collisions take time. If you stick one end of a copper rod into a flame, the other end will eventually get too hot to hold—but not instantly. Some amount of time is required for heat to be transferred from one end to the other. A process that takes place faster than the heat can be transferred is adiabatic.

NOTE You may recall reading in Chapter 18 that we are going to study only quasi-static processes, processes that proceed slowly enough to remain essentially in equilibrium at all times. Now we’re proposing to study processes that take place very rapidly. Isn’t this a contradiction? Yes, to some extent it is. What we need to establish are the appropriate time scales. How slow must a process go to be quasi-static? How fast must it go to be adiabatic? These types of calculations must be deferred to a more advanced course. It turns out—fortunately!—that many practical applications, such as the compression strokes in gasoline and diesel engines, are fast enough to be adiabatic yet slow enough to still be considered quasi-static.

For an adiabatic process, with $Q = 0$ , the first law of thermodynamics is $Δ E_{th} = W$ . Compressing a gas adiabatically $(W > 0)$ increases the thermal energy. Thus an adiabatic compression raises the temperature of a gas. A gas that expands adiabatically $(W < 0)$ gets colder as its thermal energy decreases. Thus an adiabatic expansion lowers the temperature of a gas. You can use an adiabatic process to change the gas temperature without using heat!

The work done in an adiabatic process goes entirely to changing the thermal energy of the gas. But we just found that $Δ E_{th} = n C_{V} Δ T$ for any process. Thus

W = n C_{V} Δ T (adiabatic process)

(19.32)

Equation 19.32 joins with the equations we derived earlier for the work done in isochoric, isobaric, and isothermal processes.

Gas processes can be represented as trajectories in the pV diagram. For example, a gas moves along a hyperbola during an isothermal process. How does an adiabatic process appear in a pV diagram? The result is more important than the derivation, which is a bit tedious, so we’ll begin with the answer and then, at the end of this section, show where it comes from.

First, we define the specific heat ratio $γ$ (lowercase Greek gamma) to be

γ = \frac{C_{P}}{C_{V}} = {\begin{matrix} 1.67 & monatomic gas \\ 1.40 & diatomic gas \end{matrix}

(19.33)

The specific heat ratio has many uses in thermodynamics. Notice that $γ$ is dimensionless.

An adiabatic process is one in which

p V^{γ} = constant or p_{f} V_{f}^{γ} = p_{i} V_{i}^{γ}

(19.34)

This is similar to the isothermal $p V = constant,$ but somewhat more complex due to the exponent $γ$ .

The curves found by graphing $p = constant/ V^{γ}$ are called adiabats. In FIGURE 19.20 you see that the two adiabats are steeper than the hyperbolic isotherms. An adiabatic process moves along an adiabat in the same way that an isothermal process moves along an isotherm. You can see that the temperature falls during an adiabatic expansion and rises during an adiabatic compression.

The figure illustrates a pressure versus volume graph.

FIGURE 19.20 Full Alternative Text

If we use the ideal-gas-law expression $p = n R T / V$ in the adiabatic equation $p V^{γ} = constant,$ we see that $T V^{γ - 1}$ is also constant during an adiabatic process. Thus another useful equation for adiabatic processes is

T_{f} V_{f}^{γ - 1} = T_{i} V_{i}^{γ - 1}

(19.35)

EXAMPLE 19.9 An adiabatic compression

Air containing gasoline vapor is admitted into the cylinder of an internal combustion engine at 1.00 atm pressure and $30 ° C$ . The piston rapidly compresses the gas from $500 {cm}^{3}$ to $50 {cm}^{3}$ , a compression ratio of 10.

What are the final temperature and pressure of the gas?
Show the compression process on a pV diagram.
How much work is done to compress the gas?

MODEL The compression is rapid, with insufficient time for heat to be transferred from the gas to the environment, so we will model it as an adiabatic compression. We’ll treat the gas as if it were 100% air.

SOLVE

We know the initial pressure and volume, and we know the volume after the compression. For an adiabatic process, where $p V^{γ}$ remains constant, the final pressure is

$p_{f} = p_{i} {(\frac{V_{i}}{V_{f}})}^{γ} = (1.00 atm) {(10)}^{1.40} = 25.1 atm$

Air is a mixture of $N_{2}$ and $O_{2}$ , diatomic gases, so we used $γ = 1.40$ . We can now find the temperature by using the ideal-gas law:

$T_{f} = T_{i} \frac{p_{f}}{p_{i}} \frac{V_{f}}{V_{i}} = (303 K)(25. 1) (\frac{1}{10}) = 761 K = 488 ° C$

Temperature must be in kelvins for doing gas calculations such as these.
FIGURE 19.21 shows the pV diagram. The $30 ° C$ and $488 ° C$ isotherms are included to show how the temperature changes during the process.

FIGURE 19.21 The adiabatic compression of the gas in an internal combustion engine.

FIGURE 19.21 Full Alternative Text
The work done is $W = n C_{V} Δ T$ , with $Δ T = 458 K$ . The number of moles is found from the ideal-gas law and the initial conditions:

$n = \frac{p_{i} V_{i}}{R T_{i}} = 0.0201 mol$

Thus the work done to compress the gas is

$W = n C_{V} Δ T = (0.0201 mol) (20.8 J / mol K) (458 K) = 192 J$

REVIEW The temperature rises dramatically during the compression stroke of an engine. But the higher temperature has nothing to do with heat! The temperature and thermal energy of the gas are increased not by heating the gas but by doing work on it. This is an important idea to understand.

19.7.3 Adiabatic Processes

FIGURE 19.19 The relationship of three important processes to the first law of thermodynamics.

FIGURE 19.20 An adiabatic process moves along pV curves called adiabats.

EXAMPLE 19.9 An adiabatic compression

FIGURE 19.21 The adiabatic compression of the gas in an internal combustion engine.