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## Q. 16.3

Nucleosynthesis began around the time $10^{-3}$ s, when protons and neutrons could finally remain together in the deuteron without flying apart due to the interaction of radiation. Calculate the ratio of protons to neutrons when the temperature of the universe was about $10^{10}$ K.

Strategy The ratio of protons to neutrons is purely a statistical distribution based on the available energy and the masses of the proton and neutron. The ratio is determined by the Maxwell-Boltzmann distribution from thermodynamics and the difference in masses. Because the proton has lower mass, we expect more protons to exist.

## Verified Solution

Let $\Delta m=m_{n}-m_{p}=939.566 MeV / c^{2}-$ $938.272 MeV / c^{2}=1.294 MeV / c^{2}$. The ratio of protons to neutrons is calculated to be

$\frac{\text { Number of protons }}{\text { Number of neutrons }}=\frac{e^{-m_{p} c^{2} / k T}}{e^{-m_{n} c^{2} / k T}}$

$=e^{\Delta m c^{2} / k T}=\exp \left(\frac{\Delta m c^{2}}{k T}\right)$ (16.9)

$=\exp \left(\frac{1.294 \times 10^{6} eV }{\left(8.6 \times 10^{-5} eV / K \right)\left(10^{10} K \right)}\right)$

=  4.5

As the temperature continued to decrease, the ratio of protons to neutrons continued to increase mostly due to neutron decay and somewhat to the temperature factor T in the exponential. However, the ratio of protons to neutrons bound in nuclei eventually stabilized due to the nucleosynthesis of helium.