In order for a thermonuclear fusion reaction of two deuterons (2^1H^+) to take place, the deuterons must collide and each must have a velocity of about 1 * 10^6 m/s. Find the wavelength of such a deuteron.

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Identify the formula to use: The de Broglie wavelength formula is \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) m²kg/s), \( m \) is the mass of the particle, and \( v \) is the velocity.

Determine the mass of a deuteron: A deuteron is a nucleus of deuterium, consisting of one proton and one neutron. The mass of a deuteron is approximately \( 3.34 \times 10^{-27} \) kg.

Substitute the given velocity into the formula: The velocity \( v \) is given as \( 1 \times 10^6 \) m/s.

Plug the values into the de Broglie wavelength formula: \( \lambda = \frac{6.626 \times 10^{-34}}{3.34 \times 10^{-27} \times 1 \times 10^6} \).

Simplify the expression to find the wavelength \( \lambda \).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

De Broglie Wavelength

The De Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. For a deuteron, which has mass, its momentum can be calculated using its mass and velocity, allowing us to determine its wavelength.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In the context of the deuterons in a thermonuclear fusion reaction, understanding momentum is crucial because it directly influences the De Broglie wavelength. The higher the velocity of the deuterons, the greater their momentum, which in turn affects the wavelength associated with them.

Thermonuclear Fusion

Thermonuclear fusion is a process where two light atomic nuclei, such as deuterons, combine to form a heavier nucleus, releasing a significant amount of energy. This reaction requires extremely high temperatures and velocities to overcome the electrostatic repulsion between positively charged nuclei. Understanding the conditions necessary for fusion, including the required velocities of the deuterons, is essential for analyzing the problem presented.

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