Problem 80 - University Physics with Modern Physics

Question

Paleoclimate. Climatologists can determine the past temperature of the earth by comparing the ratio of the isotope oxygen-18 to the isotope oxygen- 16 in air trapped in ancient ice sheets, such as those in Greenland. In one method for separating these isotopes, a sample containing both of them is first singly ionized (one electron is removed) and then accelerated from rest through a potential difference \(V\) . This beam then enters a magnetic field \(B\) at right angles to the field and is bent into a quarter circle. A particle detector at the end of the path measures the amount of each isotope, (a) Show that the separation \(\Delta r\) of the two isotopes at the detector is given by $$ \Delta r=\frac{\sqrt{2 e V}}{e B}\left(\sqrt{m_{18}}-\sqrt{m_{16}}\right) $$ where \(m_{16}\) and \(m_{18}\) are the masses of the two oxygen isotopes, (b) The measured masses of the two isotopes are \(2.66 \times 10^{-26} \mathrm{kg}\) \(\left(^{16} \mathrm{O}\right)\) and \(2.99 \times 10^{-25} \mathrm{kg}\) \(\left(^{18} \mathrm{O}\right)\). If the magnetic field is 0.050 T, what must be the accelerating potential \(V\) so that these two isotopes will be separated by 4.00 \(\mathrm{cm}\) at the detector?

Step-by-Step Solution

Answer

The accelerating potential \( V \) needed is approximately 1660 volts.

Step 1: Understanding the Motion of Ionized Isotopes

When isotopes are singly ionized and accelerated through a potential difference \( V \), they gain kinetic energy which can be written as \( KE = eV \), where \( e \) is the elementary charge. When these charged isotopes enter a perpendicular magnetic field \( B \), they experience a force that bends their path into a circular arc due to the Lorentz force. The radius \( r \) of their path is given by \( r = \frac{mv}{eB} \).

Step 2: Relating Velocity to Kinetic Energy

Since the isotopes are accelerated from rest, their kinetic energy \( KE = \frac{1}{2}mv^2 = eV \), allowing us to solve for \( v \). Thus, \( v = \sqrt{\frac{2eV}{m}} \). Substituting \( v \) back into the equation for \( r \) gives \( r = \frac{m}{eB} \sqrt{\frac{2eV}{m}} = \frac{\sqrt{2emV}}{eB} \).

Step 3: Deriving the Separation Formula

To find the separation \( \Delta r \) between two isotopes, calculate the difference in their path radii: \( r_{18} - r_{16} = \frac{\sqrt{2eVm_{18}}}{eB} - \frac{\sqrt{2eVm_{16}}}{eB} \). Factor out the common term \( \frac{\sqrt{2eV}}{eB} \), leading to \( \Delta r = \frac{\sqrt{2eV}}{eB} (\sqrt{m_{18}} - \sqrt{m_{16}}) \).

Step 4: Substitute Known Values for Masses and Magnetic Field

Substitute the given masses \( m_{16} = 2.66 \times 10^{-26} \) kg and \( m_{18} = 2.99 \times 10^{-25} \) kg, and \( B = 0.050 \) T into the separation formula. Set \( \Delta r = 0.04 \) m and rearrange to solve for \( V \): \( 0.04 = \frac{\sqrt{2eV}}{e \times 0.050}(\sqrt{2.99 \times 10^{-25}} - \sqrt{2.66 \times 10^{-26}}) \).

Step 5: Solving for Accelerating Potential

Rearrange the equation for \( V \) and simplify: \[ V = \left( \frac{e \times 0.050 \times 0.04}{\sqrt{2}} \right)^2 \left( \frac{1}{\sqrt{2.99 \times 10^{-25}} - \sqrt{2.66 \times 10^{-26}}} \right)^2 \]. Use \( e \approx 1.60 \times 10^{-19} \) C. Compute \( V \) using a calculator to get the desired result.

Key Concepts

Isotope Separation

Isotope separation is a critical technique used in various scientific fields, including paleoclimate analysis. It involves distinguishing and separating isotopes, which are atoms of the same element with different numbers of neutrons. In the context of studying past climates, oxygen isotopes (\(^{16}O\) and \(^{18}O\)) are especially important. By examining the ratio of these isotopes in ice cores, scientists can infer temperature changes over time.

Isotope separation can be performed through several methods, one of which includes using a magnetic field. This technique relies on the principle that isotopes, when ionized and subjected to a magnetic field, will follow different paths based on their mass. By measuring these paths, scientists can separate the isotopes and analyze the specific isotope ratios needed for climate studies. Understanding these isotopic differences is crucial for reconstructing the earth's climatic past.

• Ionization is the first step to enable isotope separation through magnetic fields.
• The mass difference in isotopes causes divergence in their paths inside a magnetic field.

Magnetic Field in Physics

A magnetic field plays a crucial role in the separation of charged particles, such as isotopes, by applying a force that alters their trajectory. When a charged particle moves through a magnetic field, it experiences a Lorentz force, which is perpendicular to both the velocity of the particle and the direction of the magnetic field.

In isotope separation, this force bends the path of the ionized isotopes into a circular trajectory. The radius of this circle is determined by the balance between the magnetic force and the centripetal force required to keep the particle moving in a curve. This radius is given by the formula \( r = \frac{mv}{eB} \), where \( m \) is the mass, \( v \) is the velocity of the particle, \( e \) is the charge, and \( B \) is the magnetic field strength.

• Magnetic fields can effectively sort isotopes by mass, allowing for detailed analysis.
• The strength and orientation of the magnetic field are critical in determining the path of isotopes.

Accelerating Potential in Charged Particles

Accelerating potential refers to the potential difference through which charged particles, like ionized isotopes, are accelerated. This potential difference imparts kinetic energy to the particles, given by the equation \( KE = eV \), where \( e \) is the electric charge, and \( V \) is the potential difference.

When isotopes are accelerated from rest, the kinetic energy they gain transforms into motion, with their velocity being expressed as \( v = \sqrt{\frac{2eV}{m}} \). Thus, accelerating potential is crucial for determining how fast isotopes will travel and how they will behave in a magnetic field.

In the context of isotope separation for paleoclimate analysis, the choice of accelerating potential affects the extent to which different isotopes can be separated when subjected to a magnetic field. The goal is to achieve a clear separation between isotopes (\(^{16}O\) and \(^{18}O\)) such that they can be detected and analyzed accurately. This separation allows scientists to make reliable inferences about past climate conditions.

• The potential difference controls the energy, and consequently, the velocity of charged particles.
• Choosing the correct accelerating potential is crucial for effective isotope separation.

Oxygen Isotopes

Oxygen isotopes, particularly \(^{16}O\) and \(^{18}O\), are vital in paleoclimatology, a discipline focused on understanding past climates. They have nearly identical chemical properties but differ slightly in mass. This difference becomes critical when studying ice cores, as the relative abundance of \(^{18}O\) to \(^{16}O\) in such samples can reveal important climate patterns.

These isotopes provide a valuable proxy for temperature estimates because water containing \(^{16}O\) evaporates more easily than water with \(^{18}O\), causing climatic variations to leave a distinct signature in ice cores. By analyzing these trends, scientists can reconstruct historical climate changes and gain insights into natural climate cycles and drivers.

The ratio of \(^{18}O\) to \(^{16}O\) is also influenced by factors such as ocean circulation and ice volume, making isotope analysis a complex but powerful tool for climatology experts. Understanding this ratio helps scientists better interpret data and refine climate models.

• Oxygen isotopes serve as indicators of historical temperature and climate variations.
• Their analysis allows scientists to infer significant climate events and trends.