OCTOBER, 1953
a WILLIAM M. SPICER and JACK HINE Georgia Institute of Technology, Atlanta, Georgia
INA widely used textbook of chemical thermodynamics there appears a statement implying that the work required to compress an ideal gas adiabatically from PI to PI is always less than that required for carrying out the analogous isothermal process. Private inquiry revealed that this opinion is shared by many students and instructors in thermodynamics, and no statement to the contrary was found in any of the textbooks available. I t is the purpose of this article to show that this is not true as a generalization, and to show further the conditions under which it is and is not correct. It will be shown that whether the isothermal or adiabatic compression requires more work, although independent of the initial temperature, depends on the ratio by which the pressure changes. At first thought it may seem reasonable that the adiabatic compression should require less work than the isothermal, since, because of the temperature increase in the adiabatic case, the final volume is greater and hence the change in volume is less than for the isothermal process. However, the amount of work done
obviously also depends upon the exact manner in which the pressure changes. Calculations show that the average pressure in the adiabatic is higher than in the corresponding isothermal compression. Thus we have two opposing factors. At lower values of P2/Pl the larger volume change accompanying the isothermal compression is the more important factor, while at higher values of Pa/P1the higher average pressure in the adibiatic compression predominates. This can be shown by Figure 1, which is a P-V diagram for the compression of one mol of a perfect monatomic gas initially a t one atmosphere and 25°C. Notice that for a final pressure of four atmospheres the volume in the adiabatic case has decreased to only 10.65 liters, whiie in the isothermal case it has decreased to 6.12 liters. Notice, however, that the average pressure is higher for the adiabatic compression. The work required, represented by the area under the curve, is 33.91 l.atm. for the isothermal and only 27.20 l.atm. for the adiabatic. On the other hand, for compression to 25 atmospheres more work is required for the adiabatic than for the isothermal, as shown by the shaded areas. The
JOURNAL OF CHEMICAL EDUCATION
502
figures are 96.30 l.atm. and 78.78 l.atm., respectively. For a gas of this type the "cross-over point" where the same amount of work is required for each type of process occurs in compression to 10.7 times the original pressnre. The value of P2/P1at the "cross-over point" varies gradually with the heat capacity of the gas, becoming less as the heat capacity increases. From the value of 10.7 for C, = 3/2 R, it decreases to 9.6 for Go = 5 / 2 R, and to 8.6 for C. = 8 / 2 R. At an infinite value of C, the two curves merge, of course, since under this condition the adiabatic process would also be isothermal.
6
5
4
S
, $3 2
I
0
1
2
3
4
5
6
7
8
9
1 0 1 1 1 2 1 3 1 4 1 5
PdP. Figure
o
4
20
16
12
24
28
V,litem risun1.
P-v cur-
for.
~ . r f e ~Gt-
I t might be instructive to calculate the work in another wav. For a reversible change in Dressure of a mol of a berfect gas, the work done by the gas in the isothermal expansion is
and in the adiabatic expansion is
Subtracting (2) from (I), we obtain
If PaP,) the right hand side of equation (4) is positive for lower values of P2/P1and negative for higher values. This is clearly shown in Figure 3, which is a plot of W I T 1 versus Pe/P1 for a monatomic gas. In comparing Figures 2 and 3, it is interesting to note that the isothermal work is similar in expansion and compression, hut there is no similarity in the adiabatic cases. The work done in an adiabatic expansion rapidly approaches a limit as P1/Pe increases. This is not surprising, since the work done in an adiabatic expansion is done entirely a t the expense of the internal energy and is therefore limited to the initial value of the internal energy of the gas (C,T,). On compression the work increases rapidly and xithout limit. 6
4
$2
I t should he pointed out that in engineering calculations of work of compression where "flow work" is taken into account, the work required for the adiabatic is always greater than that for the isothermal process, regardless of the value of PJP,.
