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Chapter 19: Problem 71

Using the kinetic molecular theory (Section 5.6), calculate the root mean square velocity and the average kinetic energy of \({ }_{1}^{2} \mathrm{H}\) nuclei at a temperature of \(4 \times 10^{7} \mathrm{~K}\). (See Exercise 46 for the appropriate mass values.)

Short Answer

Expert verified
The root-mean-square velocity of hydrogen nuclei at a temperature of \(4 \times 10^7 K\) is approximately \(6.03 \times 10^7\) m/s, and the average kinetic energy is \(8.172 \times 10^{-16} J\).

Step by step solution

01

Calculate the mass of a hydrogen nucleus

First, we need to obtain the mass of a hydrogen nucleus, which is a deuteron (2H) with one proton and one neutron. The mass of a deuteron is approximately the sum of the mass of a proton (1.6726 x 10^-27 kg) and neutron (1.6750 x 10^-27 kg). m = 1.6726 x 10^-27 kg (proton) + 1.6750 x 10^-27 kg (neutron) = 3.3476 x 10^-27 kg
02

Calculate the root-mean-square velocity

Now, we can calculate the root-mean-square velocity using the equation: v_rms = √(3kT/m) Here, k = Boltzmann's constant = 1.38 x 10^-23 J/K T = Temperature = 4 x 10^7 K m = Mass of hydrogen nucleus = 3.3476 x 10^-27 kg v_rms = √(3 * 1.38 x 10^-23 J/K * 4 x 10^7 K / 3.3476 x 10^-27 kg) v_rms = √(2.0284 x 10^4 m^2/s^2) v_rms = \(6.03 \times 10^7\) m/s The root-mean-square velocity of hydrogen nuclei is approximately \(6.03 \times 10^7\) m/s.
03

Calculate the average kinetic energy

Now we can calculate the average kinetic energy using the equation: KE_avg = (3/2) kT Here, k = Boltzmann's constant = 1.38 x 10^-23 J/K T = Temperature = 4 x 10^7 K KE_avg = (3/2) * 1.38 x 10^-23 J/K * 4 x 10^7 K KE_avg = \(8.172\) x 10^-16 J The average kinetic energy of hydrogen nuclei at a temperature of 4 x 10^7 K is approximately \(8.172\) x 10^-16 J.

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Most popular questions from this chapter

When nuclei undergo nuclear transformations, \(\gamma\) rays of characteristic frequencies are observed. How does this fact, along with other information in the chapter on nuclear stability, suggest that a quantum mechanical model may apply to the nucleus?

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