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Chapter 28: Problem 22

Radium is a radioactive element whose nucleus emits an \(\alpha\) par ticle (a helium nucleus) with a kinetic energy of about \(7.8 \times 10^{-13} \mathrm{~J}(4.9 \mathrm{MeV})\). To what amount of mass is this energy equivalent?

Short Answer

Expert verified
The energy is equivalent to a mass of \(8.67 \times 10^{-30}\) kg.

Step by step solution

01

Understanding the Energy-Mass Equivalence Formula

To find the mass equivalent of the given energy, we'll use Einstein's mass-energy equivalence formula, which is given by \(E = mc^2\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) m/s.
02

Substituting Values into the Formula

We know that the energy \(E\) is \(7.8 \times 10^{-13}\) J. We will substitute this value into the mass-energy equivalence formula to find the mass \(m\): \[7.8 \times 10^{-13} = m \times (3 \times 10^8)^2\].
03

Solving for Mass

Next, we solve the equation for \(m\). First, calculate the square of the speed of light, \((3 \times 10^8)^2 = 9 \times 10^{16}\). Then, rearrange the equation to: \[m = \frac{7.8 \times 10^{-13}}{9 \times 10^{16}}\].
04

Calculating the Mass Value

Perform the division: \(m = \frac{7.8 \times 10^{-13}}{9 \times 10^{16}} = 8.67 \times 10^{-30}\) kg. This is the mass equivalent of the given energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In our example, radium undergoes radioactive decay. This results in the emission of an \( \alpha \) particle. Over time, the unstable nuclei transform into a more stable configuration. This process releases a portion of energy.
  • The decay process is spontaneous and happens at a constant rate for a given isotope.
  • Each type of decay leads to a different kind of emitted particle, like \( \alpha \), beta (\( \beta \)), or gamma (\( \gamma \)) rays.
  • In this context, understanding the decay process helps comprehend how the radium nucleus transforms and how energy is released as the \( \alpha \) particles shoot away from the nucleus.
Breaking down the phenomenon of radioactive decay makes it easier to comprehend how energy emerges from seemingly static materials.
Alpha Particle
An alpha particle is a type of ionizing radiation ejected by the nuclei of certain radioactive elements, like radium. It is essentially a helium nucleus, consisting of two protons and two neutrons, and carries a positive charge. Alpha particles are relatively heavy and carry a significant amount of kinetic energy.
  • These particles have a relatively large mass and do not penetrate materials deeply, making them less harmful externally.
  • However, if ingested or inhaled, alpha particles can cause significant damage to living cells due to the energy they deposit locally.
  • They move at high speeds, driven by the release of energy during radioactive decay.
The understanding of alpha particles provides insight into their role and impact in nuclear reactions and radiation protection.
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion. When radium emits an \( \alpha \) particle, this particle moves with kinetic energy. In this context, it is approximately \(7.8 \times 10^{-13}\) J.
  • Kinetic energy depends on the object's mass and its velocity: \( KE = \frac{1}{2}mv^2 \).
  • For subatomic particles like the \( \alpha \) particle, the velocity can be quite high, resulting in a significant amount of energy despite their small mass.
  • This energy is a critical factor in energy-mass equivalence, linking it to changes in mass as described in Einstein's theory.
Understanding kinetic energy helps us appreciate how energy transformation occurs and why it鈥檚 vital in the study of particles and nuclear reactions.
Einstein's Mass-Energy Formula
Einstein's mass-energy formula, \( E = mc^2 \), is one of the most revolutionary equations in physics. It shows the equivalence of mass and energy, proposing that they can be converted into each other.
  • Here, \( E \) is the energy, \( m \) is the mass, and \( c \) is the speed of light (~\(3 \times 10^8\) m/s).
  • The formula reveals that a small amount of mass can convert into a vast amount of energy, as \( c^2 \) is a large number.
  • This principle is crucial in understanding phenomena such as nuclear energy and particle physics.
By applying this equation, we can compute the mass equivalent of any given amount of energy. This connection allows us to explore the fundamental workings of the universe and provides a basis for developing technologies like nuclear energy.

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Most popular questions from this chapter

Multiple-Concept Example 6 explores the approach taken in problems such as this one. Quasars are believed to be the nuclei of galaxies in the early stages of their formation. Suppose a quasar radiates electromagnetic energy at the rate of \(1.0 \times 10^{41} \mathrm{~W}\). At what rate (in \(\mathrm{kg} / \mathrm{s}\) ) is the quasar losing mass as a result of this radiation?

Which of the following quantities will two observers always measure to be the same, regardless of the relative velocity between the observers: (a) the time interval between two events; (b) the length of an object; (c) the speed of light in a vacuum; (d) the relative speed between the observers? In each case, give a reason for your answer.

The crew of a rocket that is moving away from the earth launches an escape pod, which they measure to be \(45 \mathrm{~m}\) long. The pod is launched toward the earth with a speed of \(0.55 c\) relative to the rocket. After the launch, the rocket's speed relative to the earth is \(0.75 c\). What is the length of the escape pod as determined by an observer on earth?

A woman is \(1.6 \mathrm{~m}\) tall and has a mass of \(55 \mathrm{~kg}\). She moves past an observer with the direction of the motion parallel to her height. The observer measures her relativistic momentum to have a magnitude of \(2.0 \times 10^{10} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). What does the observer measure for her height?

A woman and a man are on separate rockets, which are flying parallel to each other and have a relative speed of \(0.940 c\). The woman measures the same value for the length of her own rocket and for the length of the man's rocket. What is the ratio of the value that the man measures for the length of his own rocket to the value he measures for the length of the woman's rocket?
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