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## Q. 16.2

If the current mass of the universe consists of 75% protons and 25% ${ }^{4} He$, what is the current ratio of protons to neutrons?

Strategy We count the number of protons and neutrons in ${ }^{4} He$ and add those protons to those of single protons, from which we can find the ratio.

## Verified Solution

The mass of a proton is about 1 u, and the mass of ${ }^{4} He$ is about 4 u. If the ratio of mass of free protons to ${ }^{4} He$ is 3:1 (75% protons, 25% ${ }^{4} He$), there must be 12 free protons  for every ${ }^{4} He$. Therefore out of a total of 16 nucleons (12 free protons and 4 nucleons in ${ }^{4} He$), only 2 are neutrons, and the remaining 14 are protons (two are in ${ }^{4} He$). The ratio of protons to neutrons is about 7. This is in reasonable agreement with the fraction calculated from temperature considerations made for the time when protons and neutrons formed deuterons. When the temperature dropped below $10^{9}$ K, photons no longer had enough energy to dissociate the deuteron (2.2- MeV binding energy). After that time, free neutrons decayed (they are unstable with a half-life of 10.4 minutes), and deuterons combined to form helium.