Skip to main content
Ch 39: Wave Functions and Uncertainty
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Knight Calc 5th EditionCh 39: Wave Functions and Uncertainty문제 43b
39장, 문제 43b

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a 238U nucleus?

검증된 단계별 안내
1
Step 1: Understand the problem. The question asks for the frequency of reflections of an alpha particle within a uranium-238 nucleus. This frequency is proportional to the speed of the alpha particle and the dimensions of the nucleus. We need to calculate the frequency using the formula for frequency in terms of speed and distance.
Step 2: Identify the relevant formula. The frequency of reflections can be calculated using the formula: \( f = \frac{v}{2d} \), where \( v \) is the speed of the alpha particle and \( d \) is the approximate diameter of the nucleus. The factor of 2 accounts for the round trip of the alpha particle across the nucleus.
Step 3: Estimate the diameter of the nucleus. The diameter of a uranium-238 nucleus can be approximated using the formula for nuclear radius: \( R = R_0 A^{1/3} \), where \( R_0 \) is the nuclear radius constant (approximately \( 1.2 \times 10^{-15} \, \text{m} \)) and \( A \) is the mass number of the nucleus (238 for uranium-238). The diameter \( d \) is then \( 2R \).
Step 4: Determine the maximum speed of the alpha particle. The maximum speed can be estimated using the kinetic energy of the alpha particle, which is typically on the order of a few MeV. Use the relationship \( KE = \frac{1}{2}mv^2 \) to solve for \( v \), where \( m \) is the mass of the alpha particle (approximately \( 6.64 \times 10^{-27} \, \text{kg} \)).
Step 5: Combine the values into the frequency formula. Substitute the calculated values of \( v \) and \( d \) into \( f = \frac{v}{2d} \) to find the frequency of reflections. Ensure units are consistent throughout the calculation (e.g., meters for distance, seconds for time).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons (essentially a helium nucleus). This process reduces the atomic number of the original nucleus by two and the mass number by four, leading to the formation of a new element. Understanding alpha decay is crucial for analyzing the stability of heavy nuclei and the mechanisms of nuclear reactions.
추천 영상:
가이드 코스
04:24
Amplitude Decay in an LRC Circuit

Nuclear Potential Well

The nuclear potential well is a model that describes the forces acting within a nucleus, where nucleons (protons and neutrons) are bound together by the strong nuclear force. The potential well creates a region where particles can exist, and the energy levels within this well determine the behavior of particles, including alpha particles. The concept is essential for understanding how alpha particles can be emitted from the nucleus and the conditions that affect their stability.
추천 영상:
가이드 코스
03:55
Efficiency of a Nuclear Power Plant

Reflection Frequency

Reflection frequency refers to the rate at which an alpha particle bounces off the boundaries of the nuclear potential well. This frequency is influenced by the particle's speed and the size of the nucleus. In the context of alpha decay, a higher reflection frequency increases the likelihood of the alpha particle escaping the nucleus, making it a key factor in determining the probability of decay events.
추천 영상:
가이드 코스
06:25
Law of Reflection
관련 실천
교과서 질문

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. A 238U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a 238U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

29
views
교과서 질문

A pulse of light is created by the superposition of many waves that span the frequency range f₀ − (1/2) Δf ≤ f ≤ f₀ + (1/2) Δf, where f₀ = c/λ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs (1 fs = 1 femtosecond =10−15 s). What is the spatial length of the laser pulse as it travels through space?

51
views
교과서 질문

Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\(\psi\) (x)=\(\begin{cases}\) c\(\sqrt{1-x^{2}\)} & \(\left\)|x\(\right\)|\(\leq\) 1\(\text{ cm}\) \\ 0 & \(\left\)|x\(\right\)|\(\geq\) 1\(\text{ cm}\) \(\end{cases}\) where x is in cm. If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?

1588
views