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V1 = (Vc - m/ρ)
where ρ is the density of the oxide.
B.2.4
Decay Helium
For a radioactive species, the decay rate (and, hence, the helium generation rate, h, for alpha
decay) is
h=λN

where λ is the decay constant and N is the number of atoms of the decaying material. As a
function of time, N is given by
N = N0e-λt.

The total amount of helium generated, H, over a period of time τ is therefore
H = ∫ h dt = ∫ λ N0e-λt dt = N0(1 - e-λτ).

For values of λτ which are small, the term in parentheses can be replaced by its linear
approximation, λτ. This approximation is conservative because λτ ≥ 1 - e-λτ. Also, since the units
of H are the same as the units of N0, it is possible to consider both as moles, rather than as
atoms, and the volume of helium thus produced (in liters at STP) is
H = 22.4(1000 m/271)λτ

where m is the oxide mass in kg, and 271 is the molecular weight of PuO2.
The pressure due to this volume of helium is
PHe = (14.7)[22.4(1000 m/271)λτ](T1/273)/V1
PHe= 4.4507mλτT1/V1.

239
Pu with a half-life of 24,110 years, then
If it is assumed that the radioactive species is
Equation  becomes
PHe = 1.28x10-4 m τT1/V1.

Equation  is the same as the third term of the equation in DOE-STD-3013-99.
An alternative formulation of this term is possible. The heat generation rate of the contents is
given by
Q = EλN = EλN0e-λt

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