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Let us now consider a macroscopic body or system of bodies, and assume that the system is closed, i.e. does not interact with any other bodies. A part of the system, which is very small compared with the whole system but still macroscopic, may be imagined to be separated from the rest; clearly, when the number of particles in the whole system is sufficiently large, the number in a small part of it may still be very large. Such relatively small but still macroscopic parts will be called subsystems. A subsystem is again a mechanical system, but not a closed one; on the contrary, it interacts in various ways with the other parts of the system. Because of the very large number of degrees of freedom of the other parts, these interactions will be very complex and intricate. Thus the state of the subsystem considered will vary with time in a very complex and intricate manner.
An exact solution for the behaviour of the subsystem can be obtained only by solving the mechanical problem for the entire closed system, i.e. by setting up and solving all the differential equations of motion with given initial conditions, which, as already mentioned, is an impracticable task. Fortunately, it is just this very complicated manner of variation of the state of subsystems which, though rendering the methods of mechanics inapplicable, allows a different approach to the solution of the problem.
A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state. This may be more precisely formulated as follows. Let dp dq denote some small "volume" of the phase space of the subsystem, corresponding to coordinates q, and momenta p, lying in short intervals dq, and dp,. We can say that, in a sufficiently long time T, the extremely intricate phase trajectory passes many times through each such volume of phase space. Let dt be the part of the total time T during which the subsystem was in the given volume of phase space dp dq. When the total time T increases indefinitely, the ratio dt/T tends to some limit w = lim dt/T.
This quantity may clearly be regarded as the probability that, if the subsystem is observed at an arbitrary instant, it will be found in the given volume of phase space dpdq.
Equilibrium Statistical Mechanics and Kinetics are two branches of physical chemistry that study the behavior of molecules and atoms. The main difference between them is that Equilibrium Statistical Mechanics focuses on systems at thermodynamic equilibrium, while Kinetics studies the rates at which reactions occur.
Equilibrium and Kinetics are closely related because they both involve the study of chemical reactions and the behavior of molecules. Kinetics determines the rate at which reactions occur, while Equilibrium Statistical Mechanics looks at the distribution of molecules at equilibrium.
Equilibrium Statistical Mechanics is based on three main principles: the microscopic reversibility principle, the ergodic hypothesis, and the Boltzmann distribution. These principles help to explain the behavior of molecules in equilibrium and predict the macroscopic properties of a system.
Kinetics helps us understand chemical reactions by studying the rates at which they occur. By measuring the rate of a reaction, we can determine the order of the reaction, the rate constant, and other important parameters that describe the reaction. This information can then be used to optimize reaction conditions or predict the outcome of a reaction.
Equilibrium Statistical Mechanics and Kinetics have numerous real-life applications, such as in the design of new materials, drug development, and environmental studies. Understanding the principles of these branches of chemistry can also aid in the production of more efficient and sustainable energy sources.