Arbitrary zero of potential

Arbitrary Potential Zero The Hydrogen Scale.—Since the single electrode potential [cf. equation (10)] involves the activity of an individual ionic species, it has no strict thermodynamic significance the use of such potentials is often convenient, however, and so the difficulty is overcome by defining an arbitrary zero of potential. The definition widely adopted, following on the original proposal by Nernst, is as follows ... 

According to this definition the standard potential of the hydrogen electrode is the arbitrary zero of potential [cf. equation (7a)] electrode potentials based on this zero are thus said to refer to the hydrogen scale. Such a potential is actually the e.m.f. of a cell obtained by combining the given electrode with a standard hydrogen electrode it has, consequently, a definite thermodynamic value. For example, the potential (E) on the hydrogen scale of the electrode M, M (aM+), which is reversible with respect to the 2-valent cations M, in a solution of activity aM is the E.M.F. of the cell... 

Fortunately all practical purposes are served if we choose an arbitrary zero of potential and refer all other potentials to it. The arbitrary electrochemical zero of potential which will be adopted is that the electrode ... 

The standard potential of am electrode is defined as the standard potential of a cell in which the other (reference) electrode is the arbitrary zero of potential (equation (2)) as described above. In this chapter the methods for obtaining standard potentials from emf measurements of cells without liquid junctions will be discussed, and the available data will be used for computing such potentials. The order adopted will be, more or less, that of the increasing complexity of the methods employed, Later chapters will deal with liquid junctions and the less accurate standard potentials that can be obtained from emf values of cells containing such junctions. 

It will be recalled that the electrode H+ H2(Pt) with (H+) at unit mean ion activity is our arbitrary zero of potential. The passage of current from left to right through either cell results in the reaction... 

It is customary, but not necessary, to place the more positive electrode in the right-hand position. As we pointed out, this cell potential is always measurable as a difference in potential between two wires (for example, Pt) having the same composition. The measurement also establishes which electrode is positive relative to the other in our example, the copper is positive relative to the zinc. It does not, however, yield any hint as to the absolute value of the potential of either electrode. It is useful to establish an arbitrary zero of potential we do this by assigning the value zero to the potential of the hydrogen electrode in its standard state. 

The constcuit =9.81 m s is ccJled the acceleration of free fall. It depends on the location on the Earth s surface, but the variation is quite small. In this case, the arbitrary zero of potential energy is taken as being at the surface of the Earth (at h = 0). 

The probability density P(x) = f(x) 2 is the same for f as it is for —f the expectation values for all observable operators are the same as well. In fact, we can even multiply f by a complex number and the same result holds. The overall phase of the wavefunction is arbitrary, in the same sense that the zero of potential energy is arbitrary. Phase differences at different points in the wavefunction, on the other hand, have very important consequences as we will discuss shortly. 

Comparing (9.5) with (10.5), we see the correlation of the quantities x> and Y — W0, with the thermodynamic potentials (escaping tendencies) of the ions and electrons. The signs are naturally opposite (the arbitrary zeros of measurements do not appear), and the appearance of B and JF in (10.5) instead of k and e is due to the thermodynamic quantities being measured in moles instead of molecules. 

Also, since we are considering infinitesimal vibrational amplitudes, the terms higher than quadratic can be neglected because qt qj qt qj qt-(This approximation will be inadequate when strong anharmonicities are present.) If we choose the minimum of energy, Fo °, as the arbitrary zero of our energy scale, the potential energy, within our approximation is ... 

It is the recognised convention to take the standard hydrogen electrode, in which p(H2) = latm (101 325Nm ) and a(H ) = l, as the arbitrary zero of electrode potential. Thus H2) = 0, and, hence. 

The standard chemical potentials of the elements in the form that they are normally stable at the temperature and pressure under consideration are given an arbitrary zero... 

In the case of ions in solution, and of gases, the chemical potential will depend upon concentration and pressure, respectively. For ions in solution the standard chemical potential of the hydrogen ion, at the temperature and pressure under consideration, is given an arbitrary value of zero at a specified concentration... 

For ions in solution the standard reference state is the hydrogen ion whose standard chemical potential at = 1 is given an arbitrary value of zero. Similarly for pure hydrogen at Phj = = 0- Thus for the... 

The solution of this problem, as given by Eq. (66), must now be analyzed with consideration of the boundary conditions- at x = 0 and x = . At these two points the potential function, V(jc), becomes infinite. Therefore, for the product V(x) i (x) in Eq. (64) to remain finite at these two points, the wavefimction rfr(x) must vanish. Clearly, if rj, which is one of the arbitrary constants of integration, is chosen equal to zero in Eq. (66), the wavefunc-tion will vanish at X = 0. However, at jc = i the situation is somewhat more complicated. A little reflection will show that if the argument of the sine... 

Many scales of measurement have zero values that are arbitrary. For example, on Earth, average sea level is often assigned as the zero of altitude. In this ThoughtLab, you will investigate what happens to calculated cell potentials when the reference half-cell is changed. 

The difference between the electrostatic potential at two points is equal to the work required to bring a unit charge from one point to the other. The choice of zero potential is arbitrary, but the potential is commonly defined as zero when the particles are at infinite distance. Thus, the electrostatic potential at a point is the work required to bring a unit of charge from infinity to that point. 

Since neither of the two half-reaction potentials is known absolutely, it is necessary to propose an arbitrary /cm, relative to which all half-reaction potentials may be quoted. The half-reaction chosen to represent the arbitrary zero is the hydrogen electrode1 in which the half-reaction is the reduction of the aqueous hydrogen ion to gaseous dihydrogen ... 

Comparo Eq. (7.280) with (6.18). The potentials that are being added to and subtracted from each side here are the potentials described as constant and arising from the arbitrary designation of the potential of the reference electrode as zero (Section 63.4). 

Another potential problem is that the wave function may be contaminated not only by state s -F 1, but also states. v - - 2, s -F 3, etc. The s -F 1 annihilation operator will reduce the weights of these states in the annihilated wave function, but it will not eliminate them. Inspection of Eq. (C.33) should make clear that the higher states will contribute to the PUHF energy if they appear on both the left and right sides of the Hamiltonian expectation values with non-zero coefficients. When such contamination is important, recourse to a more complete projection operator, that annihilates an arbitrary number of spin states is available, but the computational cost increases to essentially that of an MP4 calculation. Note that the problems of the orbitals being non-ideal for the pure lowest spin state persist in this instance. 

The electrode potential is defined as the potential difference between the terminals of a cell constructed of the half-cell in question and a standard hydrogen electrode (or its equivalent) and assuming that the terminal of the latter is at zero volts. Note therefore that the electrode potential is an observable physical quantity and is unaffected by the conventions used for writing cells. The statement. . . the electrode potential of zinc is —0.76 volts. . . implies only that a voltmeter placed across the terminals of a cell consisting of standard hydrogen electrode and the zinc electrode would show this value of potential difference, with the zinc terminal negative with respect to that of the hydrogen electrode. An electrode potential is never a metal/solution potential difference , not even on some arbitrary scale. 

Each MO is characterized by a particular energy level (an eigenvalue of the MO calculation) measured on a convenient energy scale. The zero of energy is fixed at some arbitrary point, analogous to the zero point of potential (reference electrode) in an electrochemical experiment. 

For arbitrary parameters of the system, wi and ho differ dramatically from each other one of them is 1, and the other is close to zero. Within a narrow range of parameters, however, they have the same order of magnitude and one can refer to a kinetic phase transition between the two stable states it is analogous to the first-order phase transition in an equilibrium system with a potential (in the absence of quantum fluctuations) playing the role of the generalized free energy of the system [42,65,110]. This is the range of parameters that is of particular interest in the present chapter. 

To express the absolute values of single potentials is made difficult by the fact that the absolute zero electrode is not known, in respect of which other elements could be measured. It is, therefore, necessary to be satisfied with comparative values. These will be obtained by referring each potential to an exactly defined arbitrary standard electrode the potential of which is conventionally taken as zero. Such comparative potential valuos, of course, do not prevent the calculation of the EMF s of cells composed of two elements because in such instance the zero electrode potential proper appears in the corresponding equation twice once with a positive, and once with the negative sign, so being annuled in the result. 

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