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Chapter 7: Problem 51

A particle has a velocity that is \(90 . \%\) of the speed of light. If the wavelength of the particle is \(1.5 \times 10^{-15} \mathrm{~m}\), calculate the mass of the particle.

Short Answer

Expert verified
The mass of the particle is approximately \(9.09 \times 10^{-31} \mathrm{~kg}\).

Step by step solution

01

Calculate momentum using de Broglie wavelength formula

The de Broglie wavelength formula is given by: \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is the Planck's constant, and p is the momentum of the particle. We can rearrange the formula to get the momentum: \( p = \frac{h}{\lambda} \) Given the wavelength as \(1.5 \times 10^{-15}\mathrm{~m}\), Momentum, \( p = \frac{6.626 \times 10^{-34}\mathrm{~J \ s}}{1.5 \times 10^{-15}\mathrm{~m}} \)
02

Calculate the relativistic momentum

Now we know the momentum of the particle. Since we are dealing with a particle moving close to the speed of light, we will use the relativistic momentum formula: \( p = m \cdot v \cdot \frac{1}{\sqrt{1 - (\frac{v^2}{c^2})}} \), where m is the mass of the particle, v is the velocity, and c is the speed of light. The velocity is given as 90% of the speed of light: \( v = 0.9 \cdot c = 0.9 \times 3 \times 10^{8}\mathrm{~m/s} \) Now, we can plug in the values and solve for the mass: \( m = \frac{p}{v \cdot \frac{1}{\sqrt{1 - (\frac{v^2}{c^2})}}} \)
03

Simplify and solve for the mass

Plug in the values for p, v, and c into the formula above: \( m = \frac{\frac{6.626 \times 10^{-34}\mathrm{~J \ s}}{1.5 \times 10^{-15}\mathrm{~m}}}{(0.9 \times 3 \times 10^{8}\mathrm{~m/s}) \cdot \frac{1}{\sqrt{1 - (\frac{(0.9 \times 3 \times 10^{8}\mathrm{~m/s})^2}{(3 \times 10^{8}\mathrm{~m/s})^2})}}} \) Solve for m: \( m \approx 9.09 \times 10^{-31} \mathrm{~kg} \) Therefore, the mass of the particle is approximately \(9.09 \times 10^{-31} \mathrm{~kg}\).

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Most popular questions from this chapter

How many unpaired electrons are present in each of the following in the ground state: \(\mathrm{O}, \mathrm{O}^{+}, \mathrm{O}^{-}, \mathrm{Os}, \mathrm{Zr}, \mathrm{S}, \mathrm{F}, \mathrm{Ar}\) ?

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While Mendeleev predicted the existence of several undiscovered elements, he did not predict the existence of the noble gases, the lanthanides, or the actinides. Propose reasons why Mendeleev was not able to predict the existence of the noble gases.

Answer the following questions assuming that \(m_{s}\) could have three values rather than two and that the rules for \(n, \ell\), and \(m_{\ell}\) are the normal ones. a. How many electrons would an orbital be able to hold? b. How many elements would the first and second periods in the periodic table contain? c. How many elements would be contained in the first transition metal series? d. How many electrons would the set of 4 forbitals be able to hold?

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