Frank Brescia
Negative Energies
City College of New York, CUNY
New York, 10031
Many chemistry textbooks treat the theory of the H atom at more or less elementary levels. Few, however, remove the mystery of a negative energy by stressing the origin of the minus sign. None make a comparison with a missile in orbital flight. We shall also show that, in principle, a ruler can he designed to measure the potential energy of a hody as well as to measure its kinetic energy. These measuring sticks, a like many others we use, will serve to emphasize that total energy values calculated for either the missile or the H atom in various states are not absolute values. Let us first review some basic concepts. When a force is exerted in the same direction of a motion, the work done by the force, f , in moving the body in the direction r equals the gain in the kinetic energy, K, of that hody
l:
fdr = %mu,2 -
= KF- K,. = AK
(1)
in which u = velocity of the body, r, is the initial distance from the origin, and r is the distance after displacement
r increases
\
-m
Figure 1. Showing h t the direction In which the gravitational force acts is o p posite to me direction in which r increases.
freezing point of water on the Celsius temperature scale, setting the weight of a balance pan to read out zero, and taking PH,, the standard electrode potential of the hydrogen electrode as zero--the standard choice is immaterial; it is dictated by convenience. Of course, +O°K on the kinetic temperature scale is not arbitrarily chosen. Generally, the potential energy is assigned a zero value a t a specified point. For gravitational force, the accepted reference point is U,. = 0 a t r, = m when f = 0. Equation (4) then becomes
or Secondly, an increase in kinetic energy (the energy of motion) is associated with an equal decrease in potential energy,' U , (the energy of position) of a hody subject to a force, or
" = - yGMm
Illustrative is the example of a hall dropped vertically in air (air resistance neglected). As it moves downward, its speed and, therefore, its kinetic energy increases while its potential energy decreases. Since the increase in kinetic energyis given by eqn. (I),the change in potential energy is defined as
However, for short distances above the earth, it is usually assumed that the force of gravity remains constant and, for convenience, the potential energy is taken as zero a t the surface of the earth. Equation (3) then becomes
This tells us that the change in potential energy of a hody is only a function of position. I t alsosays that the change in the potential energy measures the work done by a force in arbitrarily chosen changing the position of a body from r,, (standard) reference point, t o r . For a body subject to gravitational force f=--
GMm r2
in which g is the constant of acceleration due to gravity at the earth's surface (980 cm/sec2 or 32 ft/sec2). Combined with eqn. (2). we obtain
Let U,, = 0 and r, = 0 a t the surface of the earth, then
U,= mgr
(3)
in which G is the universal gravitational constant; M is the mass of the earth, m is the mass of the body, and r, the distance measured from the center of the earth. must he eaual to or greater than R, the earth's mdius.'l'he minus sign is introduced in eon. (3) herause the fnrre of eravitsarts in a direction opposke to the direction in whici the iariahle r increases in value (see Fig. 1).Stated otherwise, the force acts downward while the direction is positive upward. The minus sign is independent of the choice made for the reference point. Then from eqns. (2) and (3) AU=U,-Uri,,=+GMm
T o evaluate eqn. (4), a choice is necessary for the reference point. As in many cases-for example, assigning O°C to the
(5)
'Provided that the work done is path independent,e.g., friction is not involved. 2Reversingthe limits an both sides of eqn. (2)
u,, - u, = -
1"
fdr
of course, also yields eqn. (5).Also, if U , = 0 at r, = R is chosen as the reference point, eqn. (4) would give u,=-7
GMm
+-GMm R
for r > R. This equation yields zero as the minimum potential energy when r s R and +GMmIR as the maximum when r = -, theopposite of eqn. (5). But the difference between any two r values remains identical. Volume 53.Number 9. September 1976 / 557
where r is the distance above the surface of the earth. mg is called the weight of a body. Thirdly, the total (mechanical) energy, E, of a body is the sum of its potential and kinetic energies3
+ Ur
E, =
(6)
Now, assume we have a missile, mass = 1.00 ton = 9.07 X
lo5 g, in circular orbit at the surface of the earth so that r =
R. Its total energy, given by eqn. (6): is the work ("escape energy", comparable to the ionization energy of the H atom) required to lift the missile from R to m, out of the sphere of influence of the earth's gravitational field. Then eqns. (5) and (6) yield
in which UR is the orbital speed of the missile relative to the earth. Equation (7) is simplified to one variable: from the fundamental equation, f = ma, where a , the acceleration of a body in circular motion, is given by u2/R so that
from which %rnuR2= GMhI2R. Then E n = -G- -M=m G M m --G M m 2R R 2R = - 6.67 X 10-8dyne em2/g2X 5.97 X 1OZ7gX 9.07 X 105g 2 x 6.37 X lo8 cm = - 2.83 X loL7erg
~
ward or downward does not change the energy difference between any two r \.slues. For example, arbitra;ily changing the scale t ~ yshifting thegraphs upward by 1lHl X 110'-ergdoes nut alter the 'escape energy" of the miasiln; '2.83X 110': erg are still reuuired to lift it from R(97.1'7 X 10"erg) t o r = (100 X 1017erg). This is comparable to changing = 0 to 10 V and so changing &Oz, = -0.763 V to +9.237 V and 6"cu = +0.346 V to 10.346 V. The emf of the cell made up of standard Cu and Zn electrodes is now 10.346 V - 9.237 V = 1.109 V, the same as 0.346 V - (-0.763 V) = 1.109 V. Also, we may arbitrarily change the sign of the E values from minus to plus so that E is interpreted as the work, instead of minus the work, required to lift the missile away from the earth's surface. Either way, the escape energy is the same. Energy differences, of course, are measurable and absolute, for exam~le. . . heat liberated (-AH).electrical work done (-AG), and ionization energy. ~ ucan t we measure the absolute values of kinetic and votential enemies of a body in motion? The answer is no. But measurements, in principle, are easilv made. Imaeine a rocket-propelled car, m = 2 X lo5 g, (road and air friction neglected) moving on a straight path. Its speed gauge has been calibrated, not in m i h , but in terms of % X car mass X u2 SO that i t s K can he directly read. Let us say its speed is 100 cmlsec so that the gauge reads 1.00 X lo9 erg. This is its kinetic energy, acceptable for all practical purposes, but actually relative to an assumed.stationary earth.5 Since gravitational force is not actingon the car, U,, = 0 and E , = 1.00 X 109erg. The car now misses a turn, goes straight off into space and is, momentarily, suspended 90 cm above the ground. Its totalenergy is now 1.00 X 109 erg U., We have a ruler especially calibrated for the mass of our car as a function of height. Where is the zero mark placed, a t or above the " eround level? Reaardless of how the ruler is held. the potential energy is always greater in a position above the eround. Plottine enerev .. notential . ... as a function of height for L o different Tern reference points. Figure 2, shows that the same result is otrtain~.dwith either ralihmrion; the potential energy of the car is 17.64 X lo9 erg so that on striking the ground i t will emit (1.00 17.64) X lo9 erg or 446 cal. The same orincinles mav he used to calculate the ionization energy of a hydrogen atom, the energy that must be supplied to move the electron to an infinite distance from the nucleus. For the electron subject to the attractive force of the (assumed) stationary nucleus, eqn. (3) becomes
eH,
Since the velocity cannot be zero or minus, the minus sign means that the potential energy of the missile is less than zero, the arbitrarily assigned value at r = m. I t therefore means that the missile is bound to the earth. Hence, work must be done to lift the missile away from the earth's surface. T o lift it out of the influence of the earth's gravitational field, +2.83 X loL7 era of work (equal to the combustion of 198 kg of Hz to H20) must be done i n the missile against the force of gravity. The changes - with distance in kinetic energy (it decreases since velocity decreases as r increases and becomes zero a t r = m ) , in potential enerw (it increases to zero at r = a ) , and in total &rev ... is absorbed) are shown -~~~~ " .lit . increases to zero since enerev in Fixure 2.'.if = O meani the missile is nut hound to theearth; it is
