Potential and Kinetic Energies

The kinetic energy, ic(J), of an object of mass m(kg) moving with velocity w(m/s) relative to the surface of the earth is 

If a fluid enters a S5 tem with a mass flow rate m(kg/s) and uniform velocity m(it /s), then 

Water flows into a process unit through a 2-cm ID pipe at a rate of 2.00 Calculate 4 for this stream in joules/second. 

SOLUTION First calculate the linear velocity (which equals the volumetric flow rate divided by the cross- 

Crude oil is pumped at a rate of 15.0 kg/s from a point 220 meters below the earth s surface to a point 20 meters above ground level. Calculate the attendant rate of increase of potential energy. 

We will assume in this book that the force depends on only a single coordinate, such as the distance between two particles, and points along that coordinate. Fortunately, this is a very common case. Then we can account for motion against a force by defining a potential energy function U r) such that the derivative of U (r) gives the force  

Note that Equation 3.6 does not use a vector symbol for the force we already know its direction from the assumption above (to within a sign). Also, the force must be conservative, which means in practice that the energy required to move from point A to point B depends only on the two positions, not on other factors such as velocity. Gravity and the force between charges are conservative friction is not. 

Equation 3.6 implies that U(r) is the negative of the antiderivative of F(r), so Equation 3.6 does not uniquely define U(r). A different potential energy function V(r ) = U(r) + C, where C is any numerical constant, would give the same force  

In other words, adding a constant C to the potential energy function offsets it, but does not change the slope (the derivative). How do we know the right value of C to use in order to get the real potential energy We don t. Forces are directly observable (they 

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To a rough approximation, the kinetic and potential energies of electrons in simple systems vary with density... 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

The typical MD Hamiltonian H of the system is the sum of kinetic and potential energy... 

I the sum of the kinetic and potential energy of an electron in the orbital lUg in the electro-atic field of the two bare nuclei. This integral can in turn be expanded by substituting the... 

The total energy of the system, called the Hamiltonian, is the sum of the kinetic and potential energies (equation 24). 

The kinetic and potential energies of the fluid at points 1 and 2 ate given by equation 103 ... 

In most apphcations to chemical processes, the kinetic- and potential-energy terms are negligible compared with the others in this event Eq. (4-359) is written... 

Calculation of Ideal Work If changes in kinetic and potential energies are neglected, Eq. (4-360) is apphcable. From the tabulated data,... 

A simplified application of the first law of thermodynamics to the air-standard Brayton cycle in Figure 2-1 (assuming no changes in kinetic and potential energy) has the following relationships ... 

If further AU = AE when the kinetic and potential energies in Equation 2.36 do not change. Equation 2.35 can be rewritten, substituting U for E, changing to the specific notation and putting the equation in differential form. 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

The Hamiltonian is made up of kinetic and potential energy terms ... 

Let us first review the Bom-Oppenheimer approximation in a bit more detail. The total Hamilton operator can be written as the kinetic and potential energies of the nuclei and electrons. 

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