Problem 17 - Physics

Question

Two isotopes of carbon, carbon- 12 and carbon- \(13,\) have masses of \(19.93 \times 10^{-27} \mathrm{~kg}\) and \(21.59 \times 10^{-27} \mathrm{~kg},\) respectively. These two isotopes are singly ionized \((+e)\) and each is given a speed of \(6.667 \times 10^{5} \mathrm{~m} / \mathrm{s}\). The ions then enter the bending region of a mass spectrometer where the magnetic field is \(0.8500 \mathrm{~T}\). Determine the spatial separation between the two isotopes after they have traveled through a half-circle.

Step-by-Step Solution

Answer

The spatial separation is approximately \(1.63 \times 10^{-2} \mathrm{~m}\).

Step 1: Identify the relevant formula

To find the spatial separation between the two isotopes, we need to use the formula for the radius of the path of a charged particle in a magnetic field. This is given by the equation \( r = \frac{mv}{qB} \), where \( r \) is the radius, \( m \) is the mass of the ion, \( v \) is its velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.

Step 2: Calculate radius for carbon-12

For carbon-12, \( m = 19.93 \times 10^{-27} \mathrm{~kg} \), \( v = 6.667 \times 10^{5} \mathrm{~m/s} \), \( q = 1.6 \times 10^{-19} \mathrm{~C} \) (charge of an electron), and \( B = 0.8500 \mathrm{~T} \). Using the formula, the radius \( r_{12} \) is:\[r_{12} = \frac{19.93 \times 10^{-27} \mathrm{~kg} \times 6.667 \times 10^{5} \mathrm{~m/s}}{1.6 \times 10^{-19} \mathrm{~C} \times 0.8500 \mathrm{~T}} \approx 9.785 \times 10^{-2} \mathrm{~m}\]

Step 3: Calculate radius for carbon-13

For carbon-13, \( m = 21.59 \times 10^{-27} \mathrm{~kg} \), and using the same values for the other constants, calculate radius \( r_{13} \):\[r_{13} = \frac{21.59 \times 10^{-27} \mathrm{~kg} \times 6.667 \times 10^{5} \mathrm{~m/s}}{1.6 \times 10^{-19} \mathrm{~C} \times 0.8500 \mathrm{~T}} \approx 1.06 \times 10^{-1} \mathrm{~m}\]

Step 4: Determine path spatial separation

The spatial separation is the difference in diameters of their semicircular paths. The diameter is twice the radius, so:\[\text{Separation} = 2(r_{13} - r_{12}) = 2(1.06 \times 10^{-1} \mathrm{~m} - 9.785 \times 10^{-2} \mathrm{~m}) \approx 1.63 \times 10^{-2} \mathrm{~m}\]

Key Concepts

Isotopes

Isotopes are variations of the same chemical element that have different numbers of neutrons but the same number of protons. This means that isotopes of an element will have the same atomic number but different mass numbers. These mass variations do not affect the chemical behavior of the element significantly but can affect its physical properties, such as mass.

• For example, carbon has isotopes such as carbon-12 and carbon-13.
• Both have 6 protons, but carbon-12 has 6 neutrons, while carbon-13 has 7 neutrons.

These small differences in mass are crucial when it comes to techniques like mass spectrometry, where the mass difference affects the path of the ions during analysis.

Magnetic Field

A magnetic field is a vector field surrounding magnetic material and moving electric charges. It exerts a force on charges and affects their motion if they are moving.

• This field is measured in Tesla (T).
• In mass spectrometry, a magnetic field is used to bend the paths of ions, causing them to travel in circular trajectories.

The force exerted by a magnetic field on a moving charged particle is perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for the circular motion of ions in devices like mass spectrometers.

Charged Particles

Charged particles are atoms or molecules that have gained or lost electrons and thus have a net electric charge. This occurs through ionization.

• Ions are the charged particles that enter a mass spectrometer for analysis.
• The carbon isotopes, for example, are ionized by losing an electron, becoming positively charged.
• The sign and magnitude of the charge affect how the particle moves in a magnetic field.

The charge of the particles determines their interaction with the magnetic field. This interaction is part of what allows mass spectrometers to differentiate isotopes based on their mass-to-charge ratio.

Radius of Path

The radius of the path describes the size of the circular trajectory a charged particle takes when moving through a magnetic field. The relationship is governed by the formula \( r = \frac{mv}{qB} \), where \( r \) is the radius, \( m \) is the mass of the ion, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.

• Heavier ions (greater mass \( m \)) will have a larger radius.
• Faster ions (greater velocity \( v \)) result in a larger radius as well.
• A stronger magnetic field \( B \) or a greater charge \( q \) reduce the radius size.

In mass spectrometry, calculating the radius helps to determine the spatial separation of isotopes, as heavier isotopes will follow larger radii than lighter ones, leading to different paths through the magnetic field.