Question 43.4: CASE STUDY Evidence in Favor of the Big Bang Model One convi......
CASE STUDY Evidence in Favor of the Big Bang Model
One convincing way to confirm the Big Bang model is by calculating the expected amount of primordial helium as a ratio N_{He} /(N_H + N_{He} ) . Helium began to form after sufficient time had passed for many neutrons to undergo beta decay. Roughly speaking, the theoretically calculated ratio of neutrons to protons was N_n /N_p ≈ 1/7 when helium began to form. Assume that all the available neutrons went into helium and the remaining protons went into hydrogen, and estimate the ratio N_{He} /(N_H + N_{He} ) . Express your answer as a percentage.
INTERPRET and ANTICIPATE
A single-helium nucleus contains two protons and two neutrons. If we assume that all the available neutrons go into helium, then an equal number of protons also go into helium. The remainder becomes hydrogen. So if the number of neutrons equaled the number of protons, there would be no hydrogen at all.
SOLVE
Every helium nucleus contains 2 neutrons, so the number of helium nuclei that can form is half the number of neutrons. Since one proton must go into helium for every neutron that does so, the number of hydrogen nuclei is the number of protons minus the number of neutrons.
Divide the numerator and denominator by N_p.
\frac{N_{ He }}{N_{ He }+N_{ H }}=\frac{\frac{1}{2}\left(N_{ n } / N_{ p }\right)}{1-\frac{1}{2}\left(N_{ n } / N_{ p }\right)}Substitute \frac{N_{ n }}{N_{ p }} \approx \frac{1}{7}.
\begin{aligned}& \frac{N_{ He }}{N_{ He }+N_{ H }}=\frac{\frac{1}{2}\left(\frac{1}{7}\right)}{1-\frac{1}{2}\left(\frac{1}{7}\right)}=\frac{1}{13} \\& \frac{N_{ He }}{N_{ He }+N_{ H }} \approx 8 \%\end{aligned}CHECK and THINK
This result is very close to the observation that about 90% of the ordinary matter in the Universe is primordial hydrogen and about 10% is primordial helium. More careful calculations have been done that include other elements created in the early Universe, and careful observations have confirmed these calculations. Although there have been great changes to our understanding of the evolution of the Universe, these fundamental ideas about how elements formed, taken from the original Big Bang theory, have changed very little and are well supported by observations.