Problem 69 - Fundamentals of Physics

Question

Atom 1 of mass \(35 \mathrm{u}\) and atom 2 of mass \(37 \mathrm{u}\) are both singly ionized with a charge of \(+e .\) After being introduced into a mass spectrometer (Fig. \(28-12\) ) and accelerated from rest through a potential difference \(V=7.3 \mathrm{kV},\) each ion follows a circular path in a uniform magnetic field of magnitude \(B=0.50 \mathrm{~T}\). What is the distance \(\Delta x\) between the points where the ions strike the detector?

Step-by-Step Solution

Answer

The separation \( \Delta x \) is given by the difference in radii of the circular paths followed by the ions in the magnetic field.

Step 1: Conceptualize the Problem

In a mass spectrometer, ions are accelerated through a potential difference and then introduced into a magnetic field, which causes them to move in circular paths. The radius of these paths depends on the ion's mass-to-charge ratio. By comparing the radii of the circular paths for the two ions, we can calculate the distance between their impact points on a detector.

Step 2: Calculate the Velocity of Ions

Using the formula for kinetic energy gained by the ions: \( KE = eV = \frac{1}{2}mv^2 \). Solve for velocity \( v \):\[ v = \sqrt{\frac{2eV}{m}} \]Given \( V = 7.3 \times 10^3 \) V and \( e = 1.6 \times 10^{-19} \) C, we calculate the velocities for the ions with masses \( m_1 = 35 \times 1.66 \times 10^{-27} \) kg and \( m_2 = 37 \times 1.66 \times 10^{-27} \) kg respectively.

Step 3: Apply Motion in Magnetic Field Formula

The radius \( r \) of an ion's circular path in a magnetic field is given by: \[ r = \frac{mv}{eB} \]Substitute the values of \( m \), \( v \), and \( B = 0.50 \) T to find the radii \( r_1 \) and \( r_2 \) for each ion.

Step 4: Calculate the Radii

For Ion 1, substitute \( m_1 \) and its calculated velocity into the radius formula: \[ r_1 = \frac{m_1 \times v_1}{eB} \]For Ion 2, substitute \( m_2 \) and its calculated velocity into the radius formula: \[ r_2 = \frac{m_2 \times v_2}{eB} \]

Step 5: Find the Difference in Radii

The distance \( \Delta x \) between the points where the ions strike the detector is the difference in their path radii:\[ \Delta x = r_2 - r_1 \]

Step 6: Insert Calculated Values

With the calculated values of \( r_1 \) and \( r_2 \), compute \( \Delta x \) to find the separation between the ions on the detector.

Key Concepts

Ion Acceleration

In a mass spectrometer, ions start from rest and are accelerated by a potential difference, which is crucial for their analysis. This ion acceleration is the process where ions gain kinetic energy as they move through a voltage. The kinetic energy gained can be expressed by the formula:

• \( KE = eV \)

This formula shows that the kinetic energy is equal to the charge of the ion \(e\) multiplied by the applied potential difference \(V\). When the ion's mass \(m\) and charge \(e\) are known, the velocity \(v\) the ion acquires can be calculated using:

• \( v = \sqrt{\frac{2eV}{m}} \)

This expression dictates that a larger potential difference or a smaller mass results in higher velocity.
Understanding how potential energy translates into kinetic energy is essential for grasping how a mass spectrometer separates ions based on their characteristics. It highlights how the initial acceleration determines the subsequent motion of ions through the instrument.

Circular Motion in Magnetic Field

Once accelerated, ions enter a magnetic field and move in circular paths. The magnetic field exerts a perpendicular force to the motion of the charged ions, changing their direction but not their speed. This specific motion is described using the concept of centripetal force, which keeps the ions moving in curved rather than straight-line motion.The radius of their circular path is determined by the balance between the magnetic force and the centripetal force, and it can be calculated with the equation:

• \( r = \frac{mv}{eB} \)

Here, \( r \) is the radius, \( m \) is the mass, \( v \) is the velocity, \( e \) is the charge, and \( B \) is the magnetic field strength.
Several factors influence the radius:

• Increased ionic mass \( m \) or velocity \( v \) leads to a larger radius.
• Stronger magnetic fields \( B \) reduce the radius of the path.

This relationship explains how ions of different masses and velocities can be separated based on their path radii in the magnetic field, forming the basis for mass spectrometry analysis.

Mass-to-Charge Ratio

The mass-to-charge ratio \(\frac{m}{e}\) plays a critical role in determining an ion's behavior in a mass spectrometer. This ratio affects how ions experience forces in electric and magnetic fields, fundamentally determining their paths and separation during analysis.

• Ions with smaller mass-to-charge ratios will follow tighter circular paths.
• Larger ratios lead to broader curves.

By measuring the radius of the circles ions make in the magnetic field, one can deduce this ratio and thus identify the ions. The formula for the radius \( r = \frac{mv}{eB} \), rearranged as \( m/e = \frac{rB}{v} \), directly relates path radius to mass-to-charge ratio.
Mass spectrometers take advantage of this differing motion to sort ions by mass, enabling highly specific analysis. Each specific mass-to-charge ratio corresponds to a unique radius in the detector, allowing for the resolution and identification of different ions.