Page 83 - Modern physical chemistry

72 The First Law for Energy

dT dV
Cv -+nR-=O, [4.49]
T V
and integrate to get

[4.50]

whence
T2 V nR1C V = 11 V1nR1CV • [4.51 ]
2
Using the ideal gas equation to eliminate the temperature leads to

[4.52]

whence
[4.53]
where
r = C p = Cv + nR = 1 + nR. [4.54]
Cv Cv Cv
In an isothermal compression of an ideal gas, product PV is constant. In a reversible
adiabatic compression, product PVY is constant. A given volume decrease thus causes a
much greater pressure rise when the process is adiabatic. Then energy that would oth-
erwise escape as heat is trapped in the system to raise the temperature and cause the
greater pressure increase.

Example 4.5

An evacuated I-liter vessel is connected by a closed tube to a I-liter vessel filled with
argon at 1 atm and 25° C. The tube is opened and the gas is allowed to reach equilibrium
at 112 atm and 25° C. What are w, fiE, and q for the overall process?
In the approximation that the vessels do not change in size, no work is done on the
system and
W=O.
Since the argon behaves as an ideal gas, its internal energy depends only on T. But since
the final temperature equals the initial temperature, we have

and
q = fiE - W = 0 - 0 = O.

The overall process is adiabatic. It is also highly irreversible. Thus it does not follow
equation (4.53).

4.9 Conditions in a Planar Pressure Pulse
The energy-capacity ratio y can be determined from the effects of adiabatic com-
pression on the given gas. In a simple experiment, one measures the speed of sound in
the gas and from this calculates y. The relationship needed can be derived simply.
Consider a steady-state planar pressure pulse in the given fluid. Observe it from a
point at rest with respect to the wave, as figure 4.4 illustrates. Locate plane 2 at a given
phase of the wave and plane 1 a given distance in front of the wave.