Mass-Energy Balance
The relationship between mass (m) and energy (E) is expressed in the following equation:
[E = mc^2 label{Eq1} ]
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where
- (c) is the speed of light ((2.998 times 10^8; m/s)), and
- (E) and (m) are expressed in units of joules and kilograms, respectively.
Albert Einstein first derived this relationship in 1905 as part of his special theory of relativity: the mass of a particle is directly proportional to its energy. Thus according to Equation (ref{Eq1}), every mass has an associated energy, and similarly, any reaction that involves a change in energy must be accompanied by a change in mass. This implies that all exothermic reactions should be accompanied by a decrease in mass, and all endothermic reactions should be accompanied by an increase in mass. Given the law of conservation of mass, how can this be true? The solution to this apparent contradiction is that chemical reactions are indeed accompanied by changes in mass, but these changes are simply too small to be detected. As you may recall, all particles exhibit wavelike behavior, but the wavelength is inversely proportional to the mass of the particle (actually, to its momentum, the product of its mass and velocity). Consequently, wavelike behavior is detectable only for particles with very small masses, such as electrons. For example, the chemical equation for the combustion of graphite to produce carbon dioxide is as follows:
[textrm{C(graphite)} + frac{1}{2}textrm O_2(textrm g)rightarrow mathrm{CO_2}(textrm g)hspace{5mm}Delta H^circ=-393.5textrm{ kJ/mol} label{Eq2} ]
Combustion reactions are typically carried out at constant pressure, and under these conditions, the heat released or absorbed is equal to ΔH. When a reaction is carried out at constant volume, the heat released or absorbed is equal to ΔE. For most chemical reactions, however, ΔE ≈ ΔH. If we rewrite Einstein’s equation as
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[Delta{E}=(Delta m)c^2 label{Eq3} ]
we can rearrange the equation to obtain the following relationship between the change in mass and the change in energy:
[Delta m=dfrac{Delta E}{c^2} label{Eq4} ]
Because 1 J = 1 (kg•m2)/s2, the change in mass is as follows:
[Delta m=dfrac{-393.5textrm{ kJ/mol}}{(2.998times10^8textrm{ m/s})^2}=dfrac{-3.935times10^5(mathrm{kgcdot m^2})/(mathrm{s^2cdot mol})}{(2.998times10^8textrm{ m/s})^2}=-4.38times10^{-12}textrm{ kg/mol} label{Eq5} ]
This is a mass change of about 3.6 × 10−10 g/g carbon that is burned, or about 100-millionths of the mass of an electron per atom of carbon. In practice, this mass change is much too small to be measured experimentally and is negligible.
In contrast, for a typical nuclear reaction, such as the radioactive decay of 14C to 14N and an electron (a β particle), there is a much larger change in mass:
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[^{14}textrm Crightarrow ,^{14}textrm N+,^0_{-1}beta label{Eq6} ]
We can use the experimentally measured masses of subatomic particles and common isotopes given in Table 20.1 to calculate the change in mass directly. The reaction involves the conversion of a neutral 14C atom to a positively charged 14N ion (with six, not seven, electrons) and a negatively charged β particle (an electron), so the mass of the products is identical to the mass of a neutral 14N atom. The total change in mass during the reaction is therefore the difference between the mass of a neutral 14N atom (14.003074 amu) and the mass of a 14C atom (14.003242 amu):
[begin{align} Delta m &= {textrm{mass}_{textrm{products}}- textrm{mass}_{textrm{reactants}}} &=14.003074textrm{ amu} – 14.003242textrm{ amu} = – 0.000168textrm{ amu}end{align} label{Eq7} ]
The difference in mass, which has been released as energy, corresponds to almost one-third of an electron. The change in mass for the decay of 1 mol of 14C is −0.000168 g = −1.68 × 10−4 g = −1.68 × 10−7 kg. Although a mass change of this magnitude may seem small, it is about 1000 times larger than the mass change for the combustion of graphite. The energy change is as follows:
[begin{align}Delta E &=(Delta m)c^2=(-1.68times10^{-7}textrm{ kg})(2.998times10^8textrm{ m/s})^2 &=-1.51times10^{10}(mathrm{kgcdot m^2})/textrm s^2=-1.51times10^{10}textrm{ J}=-1.51times10^7textrm{ kJ}end{align} label{Eq8} ]
The energy released in this nuclear reaction is more than 100,000 times greater than that of a typical chemical reaction, even though the decay of 14C is a relatively low-energy nuclear reaction.
Because the energy changes in nuclear reactions are so large, they are often expressed in kiloelectronvolts (1 keV = 103 eV), megaelectronvolts (1 MeV = 106 eV), and even gigaelectronvolts (1 GeV = 109 eV) per atom or particle. The change in energy that accompanies a nuclear reaction can be calculated from the change in mass using the relationship 1 amu = 931 MeV. The energy released by the decay of one atom of 14C is thus
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