Question

A method for determining the chemical composition of a material is Rutherford backscattering (RBS), named for the scientist who first discovered that an atom contains a high-density positively charged nucleus, rather than having positive charge distributed uniformly throughout (see Chapter 39 ). In \(\mathrm{RBS}\), alpha particles are shot straight at a target material, and the energy of the alpha particles that bounce directly back is measured. An alpha particle has a mass of \(6.65 \cdot 10^{-27} \mathrm{~kg}\). An alpha particle having an initial kinetic energy of \(2.00 \mathrm{MeV}\) collides elastically with atom \(\mathrm{X}\). If the backscattered alpha particle's kinetic energy is \(1.59 \mathrm{MeV}\), what is the mass of atom X? Assume that atom X is initially at rest. You will need to find the square root of an expression, which will result in two possible answers (if \(a=b^{2},\) then \(b=\pm \sqrt{a}\) ). Since you know that atom \(X\) is more massive than the alpha particle, you can choose the correct root accordingly. What element is atom \(\mathrm{X} ?\) (Check a periodic table of elements, where atomic mass is listed as the mass in grams of 1 mol of atoms, which is \(6.02 \cdot 10^{23}\) atoms.

Short answer

Question: Using the given data, determine the mass of atom X and identify the element. Data: Initial kinetic energy of alpha particles: 2.00 MeV Final kinetic energy of backscattered alpha particles: 1.59 MeV Mass of the alpha particle (helium nucleus): 6.65 x 10^-27 kg Answer: __(mass of atom X)__ and the element is __(element name)__.

Step-by-step solution

Calculate the initial and final velocities of the alpha particle

First, use the kinetic energy formula to calculate the initial and final velocities, \(v_{1}\) and \(v_{2}\), of the alpha particle: Initial Kinetic Energy: \(KE_1 = \frac{1}{2}mv_{1}^2 = 2.00 \text{MeV}\) Final Kinetic Energy: \(KE_2 = \frac{1}{2}mv_{2}^2 = 1.59 \text{MeV}\) Consider that 1 MeV = \(1.602 \cdot 10^{-13}\) J. Rewrite and solve for \(v_1\) and \(v_2\): \(v_1 = \sqrt{\frac{2 \cdot 2.00 \cdot 1.602 \cdot 10^{-13} \text{J}}{6.65 \cdot 10^{-27} \text{~kg}}}\) \(v_2 = \sqrt{\frac{2 \cdot 1.59 \cdot 1.602 \cdot 10^{-13} \text{J}}{6.65 \cdot 10^{-27} \text{~kg}}}\)

Find the mass of atom X using conservation of momentum and kinetic energy

Now we can use conservation of momentum and kinetic energy to find the mass of atom X. The momentum before the collision, \(p_1\), and after the collision, \(p_2\), should be equal: \(p_1 = m \cdot v_1\) \(p_2 = m \cdot v_2 + M \cdot V\) Since the collision is elastic: \(KE_1 + KE_2 = \frac{1}{2}(m Va)^2 + \frac{1}{2}(M Vb)^2\) After solving the equations, we get: \(m_2 = \frac{m^2(v_1 - v_2) }{(v_1+v_2)^2 - (v_1-v_2)^2}\) Plug in the calculated values and simplify to find the mass of atom X. Now convert the mass to number of atoms by dividing by Avogadro's number: \(M_\text{atoms} = \frac{m_\text{X}}{6.02 \cdot 10^{23}}\) Finally, identify the element corresponding to the determined atomic mass by referring to the periodic table of elements.

Key concepts

Conservation of Momentum

The principle of conservation of momentum is a fundamental concept in physics, stating that the total momentum of a closed system remains constant if no external forces are acting upon it. This Principle is critical in analyzing collisions between particles, like the one described in the Rutherford backscattering spectrometry (RBS) experiment.
In RBS, when an alpha particle collides with an atom, and both the particle and the atom are within a closed system, their total momentum before and after the collision must be equal. During the exercise, we calculate this using the formula:

• Initial momentum () = mass of alpha particle () × initial velocity ()
• Final momentum () = mass of alpha particle () × final velocity () + mass of atom X () × its velocity after the collision ()

Hence, the conservation of momentum is used to establish a relationship between the masses of the particles and their velocities to solve for unknown variables.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. In the RBS method, the collision between the alpha particle and atom X is elastic, which means that no kinetic energy is lost to other forms of energy such as heat or sound. This property allows us to write down equations for both the conservation of momentum and kinetic energy.
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The exercise employs the kinetic energy formula to determine the velocities of the particles, which play a vital role in the calculations. Subsequently, the simultaneous use of momentum and kinetic energy conservation laws enables us to find the mass of the unknown atom X.

Periodic Table of Elements

The periodic table of elements is an organized chart of chemical elements, arranged by atomic number, electron configuration, and recurring chemical properties. Elements are listed in order of increasing atomic mass, and the table is used as a predictive tool for understanding the characteristics of elements, including their reactions and compounds they are likely to form.
In the RBS exercise, once the mass of atom X is calculated, it is then converted to the number of atoms using Avogadro's number. The resulting atomic mass unit is matched to an element in the periodic table to identify atom X. The atomic mass listed on the periodic table is the mass in grams of one mole (approximately 6.02 × 10^23 atoms) of the element, which is essential for the final step of determining the identity of atom X.

Kinetic Energy Formula

The kinetic energy formula \(KE = \frac{1}{2}mv^2\) is a mathematical representation of the energy that an object possesses due to its motion. In the context of Rutherford backscattering spectrometry, kinetic energy plays a crucial role as it directly relates to the velocities of the particles involved in the collision.
During the exercise, the kinetic energy formula is used to calculate the alpha particle's initial and final velocities, which are then used in the conservation of momentum and energy equations to solve for the mass of atom X. We convert the given energy values from megaelectronvolts (MeV) to joules (J) because the kinetic energy formula requires consistency in the units of energy.