Question

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 $\textit{nuclear fusion}$, 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 $\times 10^{-15}$ 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 $\times 10^{-27}$ 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 $+2e$, and radius 1.7 $\times 10^{-15}$ 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 $\times 10^{-15}$ 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 $\frac{3}{2} kT$, where $k$ is the Boltzmann 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 $\times 10^7$ 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 77, Chapter 23: University Physics with Modern Physics
Uploaded: 2020-06-05T22:27:20-08:00
Duration: 8 min 48 s
Channel: Dading Chen

   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 $\textit{nuclear fusion}$, 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 $\times 10^{-15}$ 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 $\times 10^{-27}$ 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 $+2e$, and radius 1.7 $\times 10^{-15}$ 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 $\times 10^{-15}$ 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 $\frac{3}{2} kT$, where $k$ is the Boltzmann 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 $\times 10^7$ 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 14th Edition

Chapter 23, Problem 77

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Solved by Expert Dading Chen

06/05/2020

Step 1

The potential energy must be equal to the kinetic energy of the particles. Therefore, we have $kq^2/2r = 2 \times \frac{1}{2}mv^2$. The distance between the particles is $2r$ because each particle has a radius $r$. The reason why there is $2 \times Show more…

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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 $\textit{nuclear fusion}$, 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 $\times 10^{-15}$ 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 $\times 10^{-27}$ 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 $+2e$, and radius 1.7 $\times 10^{-15}$ 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 $\times 10^{-15}$ 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 $\frac{3}{2} kT$, where $k$ is the Boltzmann 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 $\times 10^7$ 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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Transcript

00:01 Okay, so we know due to the energy conservation law, the energy potential energy must be equal to the kinetic energy of particles.

00:08 So therefore, we have kq square over two r is equal to two times one half mv1 square.

00:14 So the distance between the particles is 2r is because the particle has a radius r.

00:19 And since there are two particles, it means distance between it is 2r, which is the diameter, okay? and the reason why is 2 times 1 half mv1 square here is because there are two particles.

00:29 And each particle has kinetic energy, which is about 1 half mv1 square.

00:34 Okay? and m is the mass of proton, and v1 is the speed of proton.

00:39 So therefore, if we do some arrangement here, we eventually have v.

00:43 V1 is equal to square root k -q squared over 2mr.

00:46 And k here is a coolon constant.

00:48 So these are the condition we know from question.

00:51 So we know the charge on the proton is 1 .6 times 10 -0 .19 coulon.

00:56 The mass of proton is 1 .67 times 10 -2 .27 kilogram.

01:00 Radius is given as 1 .2 times 10 to power negative 15 meter, and colon constant is 9 times 10 to 0 ,000 nt, times meter square per colon square.

01:08 So therefore, we can determine v1 is just simply equal to square root.

01:16 Let me see, 9 times 10 to the power of 9 newton times meter square per colon square.

01:29 And n times 1 .6 times 10 to the power negative 19, cologne square over 2 times 1 .67 times 10 to the power of negative 27 kilogram and then times 1 .2 times 10 to the power of negative 15 meter and this will give us the speed of proton is about 7 .58 times 10 to the power of 6 meter per second okay and for question b, well, we use the same formula, and this is the formula we derived from the last question.

02:26 But in this case, the case is different because the last case was a proton -proton collision.

02:34 And in question b, the case is helium -collision.

02:37 So therefore, the charge must be different, and the distance between the helium is different as well.

02:43 And the mass is different as well.

02:44 So therefore, we have b -2 is equal to square root k -q -square of md.

02:49 So we know q in this case is 2e, which is 2 times 1 .6 times 10 to 10 .190.

02:55 Which is equal to 3 .2 times 10 to 0 .190.

02:58 And the mass in this case, it was described in the question, which is 2 .99 times the mass of proton.

03:05 So we have 5 times 10 to 0 .2 .27 kilogram.

03:08 And this was given as 3 .5 times 10 to 0 .15 meter.

03:12 So therefore, we can determine v2 now...

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