Modern Engineering Thermodynamics

80 CHAPTER 3: Thermodynamic Properties
ARE “GREENHOUSE” GASES ALSO “IDEAL” GASES?
Many of the gases found in the Earth’s atmosphere behave as ideal gases, and a few are classified as “greenhouse gases.” Some
atmospheric gases trap the heat of sunlight that enters the Earth’s atmosphere just like the glass of a greenhouse traps the heat
of incoming sunlight. Many people now believe that increasing the atmospheric concentrations of these gases is producing a
global warming that will reach 3–10ºF by 2100.
Atmospheric carbon dioxide is a major greenhouse gas. Oceans and growing plants remove billions of tons of atmospheric
CO 2 from the atmosphere every year, but since the 1700s, the burning of oil, coal, and gas and continued deforestation
have increased the atmospheric CO 2 concentration by about 30%.
Carbon dioxide is used extensively in carbonated beverages. It gives the beverage its sparkle and tangy taste, and because it
forms a weak acidic solution in water (carbonic acid), it inhibits the growth of mold and bacteria. Soft drinks are carbonated
by chilling the water and cascading it in thin sheets in an enclosure containing pressurized CO 2 gas, then flavoring is added.
If the amount of CO 2 absorbed in water increases with increased surface area, then does the pressure in a soda can increase or
decrease when you shake it? Answer: The pressure actually goes down a little as you shake it because more CO 2 is dissolved
due to the increased surface area produced by the shaking. But when you open it after shaking, it squirts a lot of bubbles
because there is now too much CO 2 in solution and it comes out rapidly, as the can is depressurized when you open it.
As in the case of an incompressible substance, Eq. (3.15) gives the constant volume specific heat of an ideal gas
(since u does not depend on v) as
c v =
∂u
= du
(3.37)
∂T v dT
and if c v is constant over the temperature range from T 1 to T 2 , then integration of Eq. (3.37) gives
u 2 − u 1 = c v ðT 2 − T 1 Þ (3.38)
Thus, for a constant specific heat ideal gas, Eq. (3.38) is valid for any process (not just a constant volume process),
because the internal energy of an ideal gas does not depend on its volume. Note that, even for a constant pressure
(isobaric) process, Eq. (3.38) is valid when a constant specific heat ideal gas is used.
Combining Eqs. (3.17) and (3.35b) gives the specific enthalpy of an ideal gas as
h = u + pv = u + RT (3.39)
From Eqs. (3.19) and (3.39), we see that the constant pressure specific heat does not depend on the
pressure, so
c p =
∂h
= dh
∂T p dT = du
dT + R (3.40)
And for an ideal gas, du/dT = c v , so Eq. (3.40) becomes
c p = c v + R (3.41)
If c p is constant over the temperature range from T 1 to T 2 , integration of Eq. (3.40) gives
h 2 − h 1 = c p ðT 2 − T 1 Þ (3.42)
Thus, for a constant specific heat ideal gas, Eq. (3.42) is valid for any process (not just a constant pressure
process), because the enthalpy of an ideal gas does not depend on its pressure. Thus, even for an isochoric
(constant volume) process, Eq. (3.42) is valid when a constant specific heat ideal gas is used. Values of c p , c v ,
and the gas constant R are given in Table 3.7 for a variety of common gases at low pressure that behave as
ideal gases. A larger table can be found in Table C.13 of Thermodynamic Tables to accompany Modern Engineering
Thermodynamics.
3.9.3 Variable Specific Heats
Note that, even though the values of c p and c v for an ideal gas do not depend on p and v, they may depend on
temperature. We can improve the accuracy of an ideal gas calculation by utilizing the concept of variable specific
heats. By integrating Eqns. (3.37) and (3.40), we obtain
u 2 − u 1 =
Z T2
T 1
c v dT