Askiitians Tutor Team
Last Activity: 5 Months ago
To solve the problem regarding the ratio of singly ionized and doubly ionized positive ions in a Thomson mass spectrograph, we need to delve into the principles of how ions behave in electric and magnetic fields. The key here is to understand the relationship between the charge-to-mass ratio of the ions and the conditions under which their paths can coincide as parabolas in the spectrograph.
Understanding Ion Behavior in Fields
In a mass spectrograph, ions are accelerated by an electric field and then subjected to a magnetic field. The forces acting on the ions depend on their charge (q) and mass (m). The motion of an ion in these fields can be described by the Lorentz force equation:
F = q(E + v × B)
Where:
- F is the force acting on the ion.
- E is the electric field strength.
- v is the velocity of the ion.
- B is the magnetic field strength.
Charge-to-Mass Ratio
The charge-to-mass ratio (q/m) is crucial for determining the trajectory of the ions. For ions in a mass spectrograph, the radius of curvature of their paths in the magnetic field is given by:
r = (mv)/(qB)
From this equation, we can see that the radius of curvature is directly proportional to the mass (m) and velocity (v) of the ion, and inversely proportional to its charge (q) and the magnetic field strength (B).
Analyzing the Given Ratios
According to the problem, the ratios of electric fields (E) to magnetic fields (B) for singly ionized and doubly ionized ions are given as follows:
- Singly ionized ions: E/B = 1/2
- Doubly ionized ions: E/B = 3/2
These ratios imply that the electric field strength is half the magnetic field strength for singly ionized ions, while for doubly ionized ions, the electric field strength is 1.5 times the magnetic field strength.
Calculating the Charge-to-Mass Ratios
To find the charge-to-mass ratios for both types of ions, we can rearrange the ratios of electric and magnetic fields:
- For singly ionized ions (q = e, where e is the elementary charge):
- q/m = E/B = 1/2
- For doubly ionized ions (q = 2e):
- q/m = E/B = 3/2
Now, substituting the charge values:
- For singly ionized ions: q/m = e/m = 1/2
- For doubly ionized ions: 2e/m = 3/2
Finding the Ratio of Particles
From the equations above, we can derive the mass ratios:
- For singly ionized ions: m = 2e
- For doubly ionized ions: m = (2/3)e
Now, to find the ratio of the number of particles (N) of singly ionized ions (N1) to doubly ionized ions (N2), we can use the relationship:
N1/N2 = (mass of doubly ionized)/(mass of singly ionized)
Substituting the values:
N1/N2 = (2/3)e / (2e) = 1/3
This means that for every three doubly ionized ions, there is one singly ionized ion. Therefore, the final answer is:
Final Ratio of Particles
The ratio of singly ionized ions to doubly ionized ions is 1:3.
This analysis illustrates how the interplay of electric and magnetic fields influences the behavior of ions in a mass spectrograph, leading to the determination of their charge-to-mass ratios and ultimately the ratio of the particles themselves.
