Question

Nuclear Fusion in the Sun. The source of the sun's energy is a sequence of nuclear reactions that occur in its core. The first of these reactions involves the collision of two protons, which fuse together to form a heavier nucleus and release energy. For this process, called nuclear fusion, to occur, the two protons must first approach until their surfaces are essentially in contact. (a) Assume both protons are moving with the same speed and they collide head-on. If the radius of the proton is $1.2 \times 10^{-15} \mathrm{m}$ , what is the minimum speed that will allow fusion to occur? The charge distribution within a proton is spherically symmetric, so the electric field and potential outside a proton are the same as if it were a point charge. The mass of the proton is $1.67 \times 10^{-27} \mathrm{kg}$ . (b) Another nuclear fusion reaction that occurs in the sun's core involves a collision between two helium nuclei, each of which has 2.99 times the mass of the proton, charge $+2 e,$ and radius $1.7 \times 10^{-15} \mathrm{m}$ . Assuming the same collision geometry as in part (a), what minimum speed is required for this fusion reaction to take place if the nuclei must approach a center-to-center distance of about $3.5 \times 10^{-15} \mathrm{m} ?$ As for the proton, the charge of the helium nucleus is uniformly distributed throughout its volume. (c) In Section 18.3 it was shown that the average translational kinetic energy of a particle with mass $m$ in a gas at absolute temperature $T$ is $\frac{3}{2} k T$ , where $k$ is the Boltxmann constant (given in Appendix F) For two protons with kinetic energy equal to this average value to be able to undergo the process described in part (a), what absolute temperature is required? What absolute temperature is required for two average helium nuclei to be able to undergo the process described in part (b)? (At these temperatures, atoms are completely ionized, so nuclei and electrons move separately.) (d) The temperature in the sun's core is about $1.5 \times 10^{7} \mathrm{K}$ . How does this compare to the temperatures calculated in part (c)? How can the reactions described in parts $(a)$ and $(b)$ occur at all in the interior of the sun? (Hint ) See the discussion of the distribution of molecular speeds in Section $18.5 .$ )

Name: Problem 85, Chapter 23: University Physics with Modern Physics
Uploaded: 2020-08-29T04:52:30-08:00
Duration: 12 min
Channel: Khoobchandra Agrawal

   Nuclear Fusion in the Sun. The source of the sun's energy is a sequence of nuclear reactions that occur in its core. The first of these reactions involves the collision of two protons, which fuse together to form a heavier nucleus and release energy. For this process, called nuclear fusion, to occur, the two protons must first approach until their surfaces are essentially in contact. (a) Assume both protons are moving with the same speed and they collide head-on. If the radius of the proton is $1.2 \times 10^{-15} \mathrm{m}$ , what is the minimum speed that will allow fusion to occur? The charge distribution within a proton is spherically symmetric, so the electric field and potential outside a proton are the same as if it were a point charge. The mass of the proton is $1.67 \times 10^{-27} \mathrm{kg}$ . (b) Another nuclear fusion reaction that occurs in the sun's core involves a collision between two helium nuclei, each of which has 2.99 times the mass of the proton, charge $+2 e,$ and radius $1.7 \times 10^{-15} \mathrm{m}$ . Assuming the same collision geometry as in part (a), what minimum speed is required for this fusion reaction to take place if the nuclei must approach a center-to-center distance of about $3.5 \times 10^{-15} \mathrm{m} ?$ As for the proton, the charge of the helium nucleus is uniformly distributed throughout its volume. (c) In Section 18.3 it was shown that the average translational kinetic energy of a particle with mass $m$ in a gas at absolute temperature $T$ is $\frac{3}{2} k T$ , where $k$ is the Boltxmann constant (given in Appendix F) For two protons with kinetic energy equal to this average value to be able to undergo the process described in part (a), what absolute temperature is required? What absolute temperature is required for two average helium nuclei to be able to undergo the process described in part (b)? (At these temperatures, atoms are completely ionized, so nuclei and electrons move separately.) (d) The temperature in the sun's core is about $1.5 \times 10^{7} \mathrm{K}$ . How does this compare to the temperatures calculated in part (c)? How can the reactions described in parts $(a)$ and $(b)$ occur at all in the interior of the sun? (Hint ) See the discussion of the distribution of molecular speeds in Section $18.5 .$ )

University Physics with Modern Physics

Hugh D. Young, Roger… 12th Edition

Chapter 23, Problem 85

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Solved by Expert Khoobchandra Agrawal

08/29/2020

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The initial kinetic energy of the protons must be equal to the potential energy of interaction when fusion takes place. This gives us the equation: \[ \frac{1}{2} m v^2 + \frac{1}{2} m v^2 = \frac{k e^2}{r} \] Show more…

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Nuclear Fusion in the Sun. The source of the sun's energy is a sequence of nuclear reactions that occur in its core. The first of these reactions involves the collision of two protons, which fuse together to form a heavier nucleus and release energy. For this process, called nuclear fusion, to occur, the two protons must first approach until their surfaces are essentially in contact. (a) Assume both protons are moving with the same speed and they collide head-on. If the radius of the proton is $1.2 \times 10^{-15} \mathrm{m}$ , what is the minimum speed that will allow fusion to occur? The charge distribution within a proton is spherically symmetric, so the electric field and potential outside a proton are the same as if it were a point charge. The mass of the proton is $1.67 \times 10^{-27} \mathrm{kg}$ . (b) Another nuclear fusion reaction that occurs in the sun's core involves a collision between two helium nuclei, each of which has 2.99 times the mass of the proton, charge $+2 e,$ and radius $1.7 \times 10^{-15} \mathrm{m}$ . Assuming the same collision geometry as in part (a), what minimum speed is required for this fusion reaction to take place if the nuclei must approach a center-to-center distance of about $3.5 \times 10^{-15} \mathrm{m} ?$ As for the proton, the charge of the helium nucleus is uniformly distributed throughout its volume. (c) In Section 18.3 it was shown that the average translational kinetic energy of a particle with mass $m$ in a gas at absolute temperature $T$ is $\frac{3}{2} k T$ , where $k$ is the Boltxmann constant (given in Appendix F) For two protons with kinetic energy equal to this average value to be able to undergo the process described in part (a), what absolute temperature is required? What absolute temperature is required for two average helium nuclei to be able to undergo the process described in part (b)? (At these temperatures, atoms are completely ionized, so nuclei and electrons move separately.) (d) The temperature in the sun's core is about $1.5 \times 10^{7} \mathrm{K}$ . How does this compare to the temperatures calculated in part (c)? How can the reactions described in parts $(a)$ and $(b)$ occur at all in the interior of the sun? (Hint ) See the discussion of the distribution of molecular speeds in Section $18.5 .$ )

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Key Concepts

Nuclear Fusion

Nuclear fusion is the process in which two light atomic nuclei combine to form a heavier nucleus, releasing a significant amount of energy in the process. In stellar interiors like the Sun's core, this reaction is the fundamental mechanism powering the star, as the high temperature and pressure conditions allow charged nuclei to overcome their mutual repulsion and fuse, releasing energy according to the mass–energy equivalence principle.

Coulomb Barrier

The Coulomb barrier refers to the electrostatic repulsion between two positively charged nuclei. Before they can come close enough for the attractive nuclear force to dominate, the nuclei must have sufficient kinetic energy to overcome this repulsive potential. The magnitude of the Coulomb barrier depends on the charges and the separation distance at which the strong nuclear force begins to act, making it a critical concept in understanding the conditions required for nuclear fusion.

Kinetic Energy in Nuclear Collisions

In the context of nuclear fusion, the kinetic energy of colliding nuclei must be high enough to overcome the Coulomb barrier. This energy is derived from their velocity, and calculations often involve equating the initial kinetic energy of the nuclei to the potential energy due to their electrostatic repulsion at the point of closest approach. This concept connects the microscopic collision dynamics with macroscopic parameters such as temperature and is essential for determining the minimum conditions required for fusion.

Maxwell–Boltzmann Distribution and Thermal Energy

The Maxwell–Boltzmann distribution describes the statistical spread of particle speeds (and hence kinetic energies) in a gas at a given temperature. Because nuclear fusion reactions in stars occur in a high-temperature plasma, not all particles have the average kinetic energy; a small fraction in the high-energy tail can have energies significantly greater than the mean. This distribution is crucial for explaining how fusion reactions can occur even when the average thermal energy seems insufficient to overcome the Coulomb barrier, as these high-energy collisions are able to initiate fusion via quantum tunneling.

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Transcript

00:01 Hello everyone this is the problem based on nuclear fusion in the sun.

00:08 In the sun the energy is released due to fusion of hydrogen into helium and in the core helium into carbon.

00:23 So here we have to calculate.

00:25 I am not going to read complete paragraph this is quite lengthy.

00:30 So directly we are discussing the problem a.

00:35 Here it is asked if both the protons are moving with the same speed and they collide head on.

00:45 So this is one of the proton, this is another proton.

00:54 If they are colliding head on with same velocity, b1 is equal to b2, the radius of the proton is given 1 .2 into 10 to the power of minus 15 meters.

01:20 What is the minimum speed required for fusion to occur? so basic concept to solve this problem is conservation of energy.

01:32 When they collide, kinetic energy convert into potential energy.

01:36 So kinetic initial energy is kinetic and kinetic energy must be equal to potential energy of interaction when fusion will take place.

01:46 So it will be half mb square for one proton, half mb square for other proton and that will be k, q1, q2 ,y are potential energy.

01:57 Of interaction.

01:59 So from here we can find the velocity that will be k e squared charge on proton is plus e charge on another proton is plus e so k a square divided by m into r substitute the value you will get the answer.

02:23 K electrostatic force constant 9 into 10 to the power 9 is charge on proton m mass of proton and separation between them is 2 r 2 into 1 .2 into 10 to the power of minus 15 meter.

02:55 So on solving it, you will get the speed to be 7 .26 into 10 to the power 6 meter per second.

03:09 So this is the answer of part a.

03:15 Now we start solving the part b.

03:18 In the part b it is asked for helium helium collision.

03:23 What should be the speed? so concept to solve is same.

03:33 Helium b part for fusion of helium nucleus.

03:45 Charred on helium nucleus is plus 2e.

03:51 Mass is 4 times of that of proton that is 4 into 1 .67 into 27 k .g and the radius is given 3 .5 into 10.

04:16 10 to the power of minus 15 meters.

04:22 So concept will be the same.

04:24 If both are moving towards each other with same velocity, then kinetic energy should be equal to potential energy of interaction...

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