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The adiabatic condition of [link] can be written in terms of other pairs of thermodynamic variables by combining it with the ideal gas law. In doing this, we find that

p 1 γ T γ = constant

and

T V γ 1 = constant .

A reversible adiabatic expansion of an ideal gas is represented on the pV diagram of [link] . The slope of the curve at any point is

d p d V = d d V ( constant V γ ) = γ p V .
The figure is a plot of pressure, p on the vertical axis as a function of volume, V on the horizontal axis. Two curves are plotted. Both are monotonically decreasing and concave up.  One is slightly higher and has a greater curvature. This curve is labeled  “isothermal.” The second curve is below the isothermal curve and has  a slightly smaller curvature. This curve is labeled “adiabatic.”
Quasi-static adiabatic and isothermal expansions of an ideal gas.

The dashed curve shown on this pV diagram represents an isothermal expansion where T (and therefore pV ) is constant. The slope of this curve is useful when we consider the second law of thermodynamics in the next chapter. This slope is

d p d V = d d V n R T V = p V .

Because γ > 1 , the isothermal curve is not as steep as that for the adiabatic expansion.

Compression of an ideal gas in an automobile engine

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 × 10 5 N/m 2 , and 240 cm 3 , respectively. The mixture is then compressed adiabatically to a volume of 40 cm 3 . Note that in the actual operation of an automobile engine, the compression is not quasi-static, although we are making that assumption here. (a) What are the pressure and temperature of the mixture after the compression? (b) How much work is done by the mixture during the compression?

Strategy

Because we are modeling the process as a quasi-static adiabatic compression of an ideal gas, we have p V γ = constant and p V = n R T . The work needed can then be evaluated with W = V 1 V 2 p d V .

Solution

  1. For an adiabatic compression we have
    p 2 = p 1 ( V 1 V 2 ) γ ,

    so after the compression, the pressure of the mixture is
    p 2 = ( 1.00 × 10 5 N/m 2 ) ( 240 × 10 −6 m 3 40 × 10 −6 m 3 ) 1.40 = 1.23 × 10 6 N/m 2 .

    From the ideal gas law, the temperature of the mixture after the compression is
    T 2 = ( p 2 V 2 p 1 V 1 ) T 1 = ( 1.23 × 10 6 N/m 2 ) ( 40 × 10 −6 m 3 ) ( 1.00 × 10 5 N/m 2 ) ( 240 × 10 −6 m 3 ) · 293 K = 600 K = 328 ° C .
  2. The work done by the mixture during the compression is
    W = V 1 V 2 p d V .

    With the adiabatic condition of [link] , we may write p as K / V γ , where K = p 1 V 1 γ = p 2 V 2 γ . The work is therefore
    W = V 1 V 2 K V γ d V = K 1 γ ( 1 V 2 γ 1 1 V 1 γ 1 ) = 1 1 γ ( p 2 V 2 γ V 2 γ 1 p 1 V 1 γ V 1 γ 1 ) = 1 1 γ ( p 2 V 2 p 1 V 1 ) = 1 1 1.40 [ ( 1.23 × 10 6 N/m 2 ) ( 40 × 10 −6 m 3 ) ( 1.00 × 10 5 N/m 2 ) ( 240 × 10 −6 m 3 ) ] = −63 J .

Significance

The negative sign on the work done indicates that the piston does work on the gas-air mixture. The engine would not work if the gas-air mixture did work on the piston.

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Summary

  • A quasi-static adiabatic expansion of an ideal gas produces a steeper pV curve than that of the corresponding isotherm.
  • A realistic expansion can be adiabatic but rarely quasi-static.

Key equations

Equation of state for a closed system f ( p , V , T ) = 0
Net work for a finite change in volume W = V 1 V 2 p d V
Internal energy of a system (average total energy) E int = i ( K ¯ i + U ¯ i ) ,
Internal energy of a monatomic ideal gas E int = n N A ( 3 2 k B T ) = 3 2 n R T
First law of thermodynamics Δ E int = Q W
Molar heat capacity at constant pressure C p = C V + R
Ratio of molar heat capacities γ = C p / C V
Condition for an ideal gas in a quasi-static adiabatic process p V γ = constant

Conceptual questions

Is it possible for γ to be smaller than unity?

No, it is always greater than 1.

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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