Atom 1 of mass 35 u and atom 2 of mass 37 u are both singly ionized with a charge of +e. After b

step 1 In this question, we have two particles of M1 and M2. So this is M1. They are both positively ch

Atom 1 of mass 35 u and atom 2 of mass 37 u are both singly...

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?

Key Concepts

Relationship between Kinetic Energy, Magnetic Force, and Radius of Curvature

In a magnetic field, the radius of the circular trajectory of a charged particle is determined by balancing the magnetic Lorentz force with the required centripetal force. By equating the magnetic force (qvB) to the centripetal force (mv²/r) and using energy conservation (where half of the mass times velocity squared equals the product of the charge and potential difference), a formula is derived that relates the radius of curvature directly to the particle’s mass, charge, magnetic field strength, and the accelerating potential. This relationship is key to understanding how differences in mass result in different circular trajectories, which is fundamental to mass spectrometry.

Motion of Charged Particles in a Magnetic Field

Charged particles moving in a magnetic field experience a force perpendicular to both their velocity and the magnetic field direction, known as the Lorentz force. This force causes the ions to follow a circular or curved path when they enter a region of uniform magnetic field. The relationship governing this circular motion connects the ion's velocity, the magnetic field strength, the ion’s charge, and its mass.

Mass Spectrometry

Mass spectrometry is an analytical technique that measures the mass-to-charge ratio of ions. It involves ionizing chemical species, accelerating these ions through an electric potential, and then deflecting them in magnetic or electric fields to separate them by their masses. This method allows for the identification, quantification, and structural analysis of substances based on the trajectories of the ions produced.

Ion Acceleration through a Potential Difference

When ions are accelerated through an electric potential difference, the electric potential energy they possess is converted into kinetic energy. This process is governed by the principle of energy conservation, where the energy gained by the ion is equal to the product of its charge and the potential difference. This acceleration step is essential for imparting enough velocity to the ions so that they can be effectively manipulated and separated in subsequent magnetic fields.

Transcript

00:01 In this question, we have two particles of m1 and m2.

00:06 So this is m1.

00:10 They are both positively charged.

00:13 Then m1 is 35 u, u being the atomic mass unit.

00:18 M2 is 37 u.

00:20 They are being sent into a mass spectrometer.

00:25 So this is the region of magnetic field.

00:30 And it's going to each of them is going to do a circular motion it looks like this in a semicircle okay because of the difference in the mass they are not going to have the same radius okay and this question we want to find the delta x when they hit the plate okay so we are also given additional information such as as the accelerating voltage that the two items went through.

01:09 Okay, so the delta b is 7 .3 kilobolts, which means that they are going to have the same kinetic energy, a different speed.

01:22 Okay, then the b is given to be 0 .50 tesla.

01:26 So to find a delta x, okay, so we'll be using, okay, r equals to mv divide by bq.

01:42 So from this formula, we need to find b.

01:46 Okay, so to find b, we're using k equals to half mv squared.

01:58 And then we need this to k divide by m square.

02:07 So we want and b2.

02:10 The two particles will have different b because of different m but they have the same k...