Final answer:
The ratio of the mass of the second ion to the first ion in the mass spectrometer problem is 4. This is found by using the formula that relates the particle's mass, its charge, and the radius of its path in a magnetic field.
Step-by-step explanation:
In the context of a mass spectrometer, two ions are accelerated from rest through the same potential difference and deflected by a uniform magnetic field. When a singly ionized ion with mass m1 is deflected into a semicircle of radius R1, and a triply-ionized ion with mass m2 is deflected into a semicircle with radius 4R1, we can determine the ratio of the masses of the two ions, m2/m1, by using principles of physics regarding the motion of charged particles in magnetic fields. The centripetal force necessary to keep a charged particle moving in a circular path in the presence of a magnetic field is provided by the magnetic force. It is known that the magnetic force F is equal to qvB, where q is the charge, v is the particle's velocity, and B is the magnetic field strength. This force is also equal to the centripetal force requirement for circular motion, which is mv^2/R, where m is the mass of the particle and R is the radius of the circular path. Setting these forces equal gives us qvB = mv^2/R.By rearranging this equation and considering that the ions are accelerated through the same potential difference, we can derive that the radius R is proportional to sqrt(m/q). Since the first ion is singly ionized (q = e) and the second is triply ionized (q = 3e), and the second ion travels a path with a radius 4 times larger than the first ion (R2 = 4R1), we get R2/R1 = sqrt(m2/q2) / sqrt(m1/q1) = sqrt(m2/3e) / sqrt(m1/e) = 2sqrt(m2/m1). Because R2/R1 = 4, we can write 4 = 2sqrt(m2/m1). Upon squaring both sides, we get 16 = 4m2/m1, which results in m2/m1 = 4. So, the ratio of the masses is 4.