The coulomb barrier between two protons is about 400 keV. Setting \( \frac{3}{2}kT \) equal to this barrier gives a temperature of about \( 3\times10^{9}\,\mathrm{K} \), yet hydrogen fuses in the core of the sun, which is only at about \( 1.5\times10^{7}\,\mathrm{K} \). This is possible because
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → some protons in the core have kinetic energies much above the average energy
Two protons can fuse only when they come close enough for the short-range nuclear force to act. For this they need enough kinetic energy to overcome the coulomb barrier between them.
The quantity \( \frac{3}{2}kT \) is only the average kinetic energy of a proton at temperature \( T \). So \( 3\times10^{9}\,\mathrm{K} \) is the temperature at which an average proton could cross the barrier.
In a hot gas the protons have energies spread well above and below this average.
Option (1) the gravitational pull in the core overcomes the coulomb repulsion between the protons — Incorrect. Gravity keeps the core of the sun hot and dense. But the gravitational force is much weaker than the Coulomb force, so it cannot act against the repulsion between two protons.
Option (2) the nuclear force between two protons acts over a much larger distance at such high temperatures — Incorrect. The nuclear force is short-ranged at any temperature. It falls rapidly to zero once the separation is more than a few femtometres.
Option (3) some protons in the core have kinetic energies much above the average energy — Correct. Protons of average energy cannot cross the barrier at \( 1.5\times10^{7}\,\mathrm{K} \). But a few protons have energies far above the average. These protons can cross it, and they are the ones that fuse.
Option (4) the fusion in the core takes place between neutrons, which feel no coulomb repulsion — Incorrect. The fuel in the core is hydrogen. The first step of the proton-proton cycle joins two protons, \( {}^{1}_{1}\mathrm{H}+{}^{1}_{1}\mathrm{H}\rightarrow{}^{2}_{1}\mathrm{H}+e^{+}+\nu \), and both particles are positively charged.
NCERT (2026-27):
Nuclear Fusion – Energy Generation in Stars – “Thermonuclear fusion is the source of energy output in the interior of stars. The interior of the sun has a temperature of 1.5×107 K, which is considerably less than the estimated temperature required for fusion of particles of average energy. Clearly, fusion in the sun involves protons whose energies are much above the average energy.”