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Chapter 40: Problem 51

What is the average kinetic energy of protons at the center of a star where the temperature is \(1.00 \cdot 10^{7} \mathrm{~K} ?\) What is the average velocity of those protons?

Short Answer

Expert verified
Answer: The average kinetic energy of protons is \(2.07 \cdot 10^{-16} \mathrm{~J}\), and their average velocity is \(4.98 \cdot 10^5 \mathrm{~m/s}\).

Step by step solution

01

Calculate the average kinetic energy of protons

Using the Equipartition theorem, the average kinetic energy per particle is given by: \(KE = \frac{3}{2}kT\) Given temperature \(T = 1.00 \cdot 10^{7}\mathrm{~K}\) and Boltzmann constant \(k = 1.38 \cdot 10^{-23} \mathrm{~J / K}\), we can find the average kinetic energy as: \(KE = \frac{3}{2} \times 1.38 \cdot 10^{-23} \mathrm{~J / K} \times 1.00 \cdot 10^{7} \mathrm{~K} = 2.07 \cdot 10^{-16} \mathrm{~J}\)
02

Calculate the average velocity of protons

Now that we have the average kinetic energy, we can use the kinetic energy formula \(\frac{1}{2}mv^2\) to find the average velocity of protons: \(2.07 \cdot 10^{-16} \mathrm{~J} = \frac{1}{2}(1.67 \cdot 10^{-27} \mathrm{~kg})(v^2)\) Solving for \(v\), we get: \(v^2 = \frac{2(2.07 \cdot 10^{-16} \mathrm{~J})}{1.67 \cdot 10^{-27} \mathrm{~kg}}\) \(v^2 = 2.48 \cdot 10^{11} \mathrm{~m^2/s^2}\) \(v = \sqrt{2.48 \cdot 10^{11} \mathrm{~m^2/s^2}}\) \(v = 4.98 \cdot 10^5 \mathrm{~m/s}\) So, the average velocity of protons at the center of the star is \(4.98 \cdot 10^5 \mathrm{~m/s}\).

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Most popular questions from this chapter

\(^{214} \mathrm{Pb}\) has a half-life of \(26.8 \mathrm{~min}\). How many minutes must elapse for \(90.0 \%\) of a given sample of \({ }^{214} \mathrm{~Pb}\) atoms to decay?

A certain radioactive isotope decays to one-eighth its original amount in \(5.0 \mathrm{~h}\). a) What is its half-life? b) What is its mean lifetime?

In a simple case of chain radioactive decay, a parent radioactive species of nuclei, A, decays with a decay constant \(\lambda_{1}\) into a daughter radioactive species of nuclei, B, which then decays with a decay constant \(\lambda_{2}\) to a stable element C. a) Write the equations describing the number of nuclei in each of the three species as a function of time, and derive an expression for the number of daughter nuclei, \(N_{2}\), as a function of time, and for the activity of the daughter nuclei, \(A_{2},\) as a function of time. b) Discuss the results in the case when \(\lambda_{2}>\lambda_{1}\left(\lambda_{2} \approx 10 \lambda_{1}\right)\) and when \(\lambda_{2}>>\lambda_{1}\left(\lambda_{2} \approx 100 \lambda_{1}\right)\).

Pu decays with a half-life of 24,100 yr via a 5.25 \(\mathrm{MeV}\) alpha particle. If you have a \(1.00 \mathrm{~kg}\) spherical sample of \({ }^{239} \mathrm{Pu},\) find the initial activity in \(\mathrm{Bq} .\)

The radon isotope \({ }^{222} \mathrm{Rn}\), which has a half-life of 3.825 days, is used for medical purposes such as radiotherapy. How long does it take until \({ }^{222} \mathrm{Rn}\) decays to \(10.00 \%\) of its initial quantity?
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