Modern Thermodynamics
- Chapter 1
For a mixture of ideal gases, we have the Dalton's law of partial pressures according to which
the pressure exerted by each component of the mixture is independent of the other components of the
mixture, and each component obeys the ideal gas equation. Thus, if pk is the partial pressure due to
component k, we have:
pkV = NkRT
Joseph Louis Gay-Lussac (1778-1850), who made important contributions to the laws of gases,
discovered that a dilute gas expanding into vacuum did so without change in temperature. James Prescott
Joule(1818-1889) also verified this fact in his series of experiments that established the equivalence
between mechanical energy and heat. In chapter 2 we will discuss Joule's work and the law of
conservation of energy in detail. When the concept of energy and its conservation was established, the
implication of this observation became clear. Since a gas expanding into vacuum does not do any work
during the processes of expansion, its energy does not change. The fact that temperature does not change
during expansion into vacuum while the volume and pressure change, implies energy of a given amount
of ideal gas depends only on its temperature, T, but not on its volume or pressure. Also, a change in the
ideals gas temperature occurs only when its energy is changed through exchange of heat or mechanical
work. These observations lead to the conclusion that energy of a given amount of ideal gas is a function
only of temperature T. Since the amount of energy(heat) needed to increase the temperature of an ideal
gas is proportional to the amount of the gas, the energy is proportional N, the amount of gas in moles.
Thus, the energy of the ideal gas, U(T,N), is a function only of the temperature T and the amount of gas,
N. It can be written as:
U(T,N) = NUm (T)
In which Um is the total internal energy per mole, or molar energy. For a mixture of gases the total
energy is the sum of the energies of the components:
U(T,N) = (T,Nk )
in which the components are indexed by k. Later developments established that:
Um = cRT + U0
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