Stability of Single-Phase Substance

Let the temperature, , and the volume, , be the two independent parameters that specify the macrostate of . Consider, first, a situation where is fixed, but is allowed to vary. Suppose that the minimum of occurs for , when . Expanding about its minimum, and writing , we obtain

(9.31) |

Here, all derivatives are evaluated at . Because is a stationary point (i.e., a maximum or a minimum), to first order in . In other words,

Furthermore, because is a minimum, rather than a maximum, to second order in . In other words,

According to Equation (9.30), the condition, (9.32), that be stationary yields

However, the fundamental thermodynamic relation

implies that if is fixed then

(9.36) |

Thus, Equation (9.34) becomes

Recalling that the derivatives are evaluated at , we get

(9.38) |

Thus, we arrive at the rather obvious conclusion that a necessary condition for equilibrium is that the temperature of the subsystem be the same as that of the surrounding medium.

Equations (9.33) and (9.37) imply that

(9.39) |

Recalling that the derivatives are evaluated at , and that , the second term on the right-hand side of the previous equation vanishes, and we obtain

However, this derivative is simply the heat capacity, , at constant volume. Thus, we deduce that a fundamental criterion for the stability of any phase is

(9.41) |

The previous condition is a particular example of a very general rule, known as *Le Chátelier's principle*, which states that

If a system is in a stable equilibrium then any spontaneous change of its parameters must give rise to processes that tend to restore the system to equilibrium.To show how this principle applies to the present case, suppose that the temperature, , of the subsystem has increased above that of its surroundings, , as a result of a spontaneous fluctuation. The process brought into play is the spontaneous flow of heat from system , at the higher temperature, to its surroundings , which brings about a decrease in the energy, , of (i.e., ). By Le Chátelier's principle, this energy decrease must act to decrease the temperature of (i.e., ), so as to restore to its equilibrium state (in which its temperature is the same as its surroundings). Hence, it follows that and have the same sign, so that , in agreement with Equation (9.40).

Suppose, now, that the temperature of subsystem is fixed at , but that its volume, , is allowed to vary. We can write

where , and the expansion is about the volume at which attains its minimum value. The condition that is stationary yields

The condition that the stationary point is a minimum gives

Making use of Equation (9.30), we get

However, Equation (9.35) implies that

(9.46) |

Hence,

(9.47) |

or

because . Thus, the condition (9.43) yields

(9.49) |

This rather obvious result states that in equilibrium the pressure of subsystem must be equal to that of the surrounding medium.

Equations (9.44) and (9.48) imply that

When expressed in terms of the isothermal compressibility,

the stability criterion (9.50) is equivalent to

(9.52) |

To show how the preceding stability condition is consistent with Le Chátelier's principle, suppose that the volume of subsystem has increased by an amount , as a result of a spontaneous fluctuation. The pressure, , of must then decrease below that of its surroundings (i.e., ) in order to ensure that the net force exerted on by its surroundings is such as to reduce the volume of back towards its equilibrium value. In other words, and must have opposite signs, implying that .

The preceding discussion allows us to characterize the spontaneous volume fluctuations of the small subsystem . The most probable situation is that where takes the value that minimizes , so that . Let denote the probability that the volume of lies between and . According to Equation (9.29),

However, when is small, the expansion (9.42) is applicable. By virtue of Equations (9.43), (9.50), and (9.51), the expansion yields

(9.54) |

Thus, Equation (9.53) becomes

where we have absorbed into the constant of proportionality, . Of course, can be determined via the requirement that the integral of over all possible values of the volume, , be equal to unity.

The probability distribution (9.55) is a Gaussian whose maximum lies at . Thus, is equal to the mean volume, . Moreover, the standard deviation of the spontaneous fluctuations of about its mean value is

(9.56) |

(See Section 2.9.) The fluctuations in imply corresponding fluctuations in the number of particles per unit volume, , (and, hence, in the mean density) of the subsystem . The fluctuations in are centered on the mean value . For relatively small fluctuations, . Hence,

(9.57) |

Note that the relative magnitudes of the density and volume fluctuations are inversely proportional to the square-root of the volume, , under consideration.

An interesting situation arises when

In this limit, , and the density fluctuations become very large. [They do not, however, become infinite, because the neglect of third-order, and higher, terms in the expansion (9.42) is no longer justified.] The condition defines the so-called