Point-mass particle

A model system that represents a dilute gas consists of randomly moving noninteracting point-mass particles that obey classical mechanics. 

The particles are vanishingly small in size. We call them point-mass particles. 

According to classical mechanics, the state of a point-mass particle is specified by its position and its velocity. If the particle moves in three dimensions we can specify its position by the three Cartesian coordinates x, y, and z. These three coordinates are equivalent to a three-dimensional vector, which we denote by r and call the position vector. This vector is a directed line segment that reaches from the origin of coordinates to the location of the particle. We call x, y, and z the Cartesian components of the position vector. We will denote a vector by a letter in boldface type, but it can also be denoted by a letter with an arrow above it, as in T, by a letter with a wavy underscore, as in r, or by its three Cartesian components listed inside parentheses,... 

We now want to show that our model gas of point-mass particles obeys the ideal gas law. We assume that the box confining our model system is rectangular with walls that are perpendicular to the coordinate axes and that the walls are smooth, slick, flat, and impenetrable. A collision of a molecule with such a wall is called a specular collision, which means (1) It is elastic. That is, the kinetic energy of the molecule is the same before and after the collision. (2) The only force exerted on the particle is perpendicular to the wall. 

Now consider a model system that contains a mixture of several gaseous substances, each one represented by point-mass particles. We denote the number of substances by c. We continue to assume that the molecules have zero size and do not exert forces on each other, so the molecules of each substance will move just as though the other substances were not present. Because the pressure is the sum of the effects of individual molecules, the total pressure is the sum of the pressures exerted by each set of molecules ... 

If the force on a particle is a known function of position, Eq. (E-1) is an equation of motion, which determines the particle s position and velocity for all values of the time if the position and velocity are known for a single time. Classical mechanics is thus said to be deterministic. The state of a system in classical mechanics is specified by giving the position and velocity of every particle in the system. All mechanical quantities such as kinetic energy and potential energy have values that are determined by the values of these coordinates and velocities, and are mechanical state functions. The kinetic energy of a point-mass particle is a state function that depends on its velocity ... 

In Section 15.3 we discussed a model system consisting of a single point-mass particle absolutely confined in a three-dimensional rectangular box. We now go through the mathematics of the Schrodinger equation for this system. We choose the potential energy function ... 

At this point, we shall restrict attention to the mechanics of systems of point-mass particles. This means that particles are taken to have no volume. The mass of each particle is treated as if enclosed in an infinitesimally small region of space, a point. And mostly, we will restrict our attention to conservative systems, which are those for which the potential energy has no explicit dependence on time (i.e., there are no potentials changing with time). These restrictions are not an aspect of the particular mechanical formulation they are used here as a convenience that is in keeping with the types of systems of most inune-diate interest in chemistry. 

Write the potential energy of the following system of point-mass particles and harmonic springs as a function of the position coordinates of the particles. Assume that they are constrained to move only in the x-y plane. 

For a system of several point-mass particles, the total angular momentum is the vector sum of the angular momenta of each of the particles. 

Equation (1.3) provides abridge between the observable macroscopic states and the microscopic states of any system. If there were a way to know the microscopic state of the system at different times then all thermodynamic properties could be determined. Assuming a classical system of point-mass particles, Newtonian mechanics provides such a way. We can write Newton s second law for each particle i as follows ... 

Consider a system with N interacting classical point-mass particles in constant motion. These are physical bodies whose dimensions can be assumed immaterial to their motion. Consider the particle positions LvL2 > Liv> where r, = (jc, y, z,), for all /, in a Cartesian coordinate system. 

Consider a rigid rotor consisting of two point mass particles of mass m joined together at a distance r. Express the kinetic energy of the rigid rotor in terms of... 

Thus far, we have only introduced systems of monoatomic, or point mass, particles. In this chapter we present a statistical mechanical derivation of polyatomic system thermodynamics. We focus on diatomic molecules in an ideal gas phase and we use the canonical ensemble partition function. The reason this ensemble is chosen is the ease with which the integration of the Boltzmann factor can be performed over the entire phase space. All the ensembles of molecular systems are again equivalent at the thermodynamic limit. 

Obtain Moldyn from the textbook website (http //statthermo.sourceforge.net/). Moldyn is a set of codes which simulates point mass particles that interact with Lennard-Jones forces. Moldyn uses the velocity-Verlet integration algorithm in the NVE ensemble. Read the README file on how to run the programs and do this homework problem. 

Procedure Fill a %x4 incb test tube to a depth of 1 inch with the specimen ground to pass a No 16 mesh screen. Submerge the tube to a depth of 2 inches in an oil bath previously heated to 120°, place a thermometer in the EtCell and increase the bath temp at the rate of 2°C per min, while stirring the specimen by means of thermometer. At a temp approx 25° below fusion point the particles soften and tend to cohere, making it difficult to stir. Continue beating and working the mass until it fuses and becomes plastic. Read the "thermoplastic point" as the temp at which the plastic mass can be repeatedly drawn into thread by pulling the thermometer from the mass. Duplicate detns should check within 3°C... 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

In summary, for each trap or standing-crop sampling point, eight particle-size fractions were created with nominal cutoffs of 508, 212, 114, 63, 19, 8.2, 1.0, and 0.4 pm. The 600 standing-crop and 1000 trap-mass fractions were chemically analyzed for phosphorus and major and trace elemental composition. Major particle types were identified and enumerated by optical microscopy. 

Although Eq. (110) is a classical result, Eq. (115) is decoherence protected for zero rest mass particles since dJ fconj is nondiagonal with a zero determinant (for all values of r). For a particle with mo 0, the only singular point is at r = RLS. This occurs when Newton s law of gravity is appended with an appropriate boundary condition, see Eqs. (98)—(101). Nevertheless, the introduction of the operator Es replacing the conventional notion of a rest mass implies the construction of an invariant d5 onj = -c2ds2, which by definition must be zero for photons. The result is the well-known line element expression (in the spherical case)... 

Instead of using the virtual impactor approach, North American air monitoring programs in the 1980s and later have adopted simpler reference methods that use the weighing of filters in the laboratory. The filters are obtained from samplers equipped with an inlet device that provides for a sharp cut-point in particle entry for samples of particles < 10 xm diameter or <2.5 [im diameter, which are operated over a fixed time period of 24 hours. The inlet fractionation is facilitated either by a carefully designed cyclone or by an impactor. The combination of the two samplers can give estimates of mass concentration for fine-particle and coarse-particle concentrations. 

Particles can be regarded as point masses which exert no force on each other beyond a separation distance r > R (R is known as the interactive radius). At r < R there is an infinite repulsive force between the particles involved. They therefore behave like perfectly elastic spheres of radius R/2. 

Surface area is controlled independently of either particle size or structure. If the reaction were quenched at the end of its primary reaction time (as mentioned, at the point of particle size and structure formation, about 1.2 to 2 msec.), the surface area for this particular black would be at its lowest. If, on the other hand, the reaction mass is allowed to remain at temperature for a period of 8 to 10 times the initial period, i.e., for 12 to 20 msec., additional reactions continue to occur. Although the reaction for the formation of carbon black has been completed, further reactions between carbon, carbon dioxide, and water result in an increase in surface area. These reactions are slow compared to the formation of carbon reactions and, allowing these reactions to proceed for about 20 msec., the effect on particle size and... 

Consider a one-dimensional (usually almost infinite) set of N atoms, molecules, or point masses, all equally spaced at inter-particle distances d along the real-space coordinate x, with Bom-von Karman periodic boundary conditions for the potential energy ... 

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