The neutron generation time ${\mathit{t}}_{\mathrm{gen}}$(see Problem 19) in a particular reactor is $\mathbf{\text{1.0ms}}$. If the reactor is operating at a power level of $\mathbf{\text{500MW}}$, about how many free neutrons are present in the reactor at any moment?

• The neutron generation time in the reactor,$t_{gen}=1.0\text{ms}={\text{10}}^{-3}\text{s}$
• The operating power level of the reactor, $P=500\text{MW}=\text{500}\times {\text{10}}^6\text{W}$

Neutron generators are neutron source devices that contain compact linear particle accelerators and that produce neutrons by fusing isotopes of hydrogen together. In a nuclear reactor, there occur many fission reactions due to the number of neutron generations taking each moment due to its operating power. Thus, the number of neutrons is given by the division of the total heat by the reactor to the heat by every fission reaction.

Formulae:

The power level of a reactor is,

$P=\frac{Q}{t}$ ..... (i)

Where, $Q$is the heat generated by the reactor and $t$is the generation time of the process.

The number of particle generated in a reactor,

$N=\frac{Q_{total}}{Q}$ ..... (ii)

Where, $Q_{total}$is the amount of heat generated by the reactor and $Q$is the amount of heat generated by a fission reaction.

In every fission reaction, the amount of heat generated is given by:

$\begin{array}{rcl}Q& =& 200\text{MeV}\\ & =& (200\times {10}^6\times 1.6\times {10}^{-19})\text{J}\\ & =& \text{3.2}\times {10}^{-11}\text{J}\end{array}$

As many number of fission reactions undergo within the reactor, the total amount of heat generated by the reactor is given by the power of the reactor within the generation time of the neutrons,

Now, using the given data in equation (i), the amount of heat generated by the reaction due to its operating power can be given as follows:

$\begin{array}{rcl}{Q}_{total}& =& Pt\\ & =& (500\times {10}^6\text{W})\left({\text{10}}^{-3}\text{s}\right)\\ & =& 5\times {10}^5\text{J}\end{array}$

Thus, the number of free neutrons generated in the reaction can be given using the given data in equation (ii) as follows:

$\begin{array}{rcl}N& =& \frac{5\times {10}^5\text{J}}{3.2\times {10}^{-11}\text{J}}\\ & =& 1.6\times {10}^{16}\end{array}$

Hence, $1.6\times {10}^{16}$neutrons are present in the reactor.