In one type of mass spectrometer, ions having the same velocity move through a uniform magnetic field. The spectrometer is being used to distinguish 12C+ and 14C+ ions that have the same charge. The 12C+ ions move in a circle of diameter 25 cm. (a) What is the diameter of the orbit of 14C+ ions? (b) What is the ratio of the frequencies of revolution for the two types of ion?

To solve this problem, we need to use the principles of circular motion and the relationship between the speed of an ion and the radius of its orbit in a magnetic field.

(a) To find the diameter of the orbit for 14C+ ions, we can use the fact that the ratio of the diameters is equal to the ratio of the masses. Since 12C+ and 14C+ have the same charge, the ratio of the masses is simply the ratio of their isotopic masses, which is 14/12.

Let's denote the diameter of the orbit for 14C+ ions as D. Since the diameter of the orbit for 12C+ ions is given as 25 cm, we can set up the following equation to find D:

(Diameter of 14C+ orbit) / (Diameter of 12C+ orbit) = (Mass of 14C+ ion) / (Mass of 12C+ ion)

Therefore:

D / 25 cm = 14 / 12

Cross-multiplying:

D = 25 cm * 14 / 12

Simplifying:

D = 35 cm

Therefore, the diameter of the orbit for 14C+ ions is 35 cm.

(b) The frequency of revolution for an ion in a magnetic field is directly proportional to its speed, which can be calculated using the formula:

Speed = (Charge of the ion * Magnetic field strength) / (Mass of the ion)

Since the charge and magnetic field strength are the same for both 12C+ and 14C+ ions, we can simplify the ratio of the frequencies of revolution to the ratio of their speeds, which is in turn equal to the ratio of their masses.

The ratio of the frequencies of revolution for the two types of ions is:

(Frequency of 14C+ ion) / (Frequency of 12C+ ion) = (Mass of 14C+ ion) / (Mass of 12C+ ion)

Therefore:

(Frequency of 14C+ ion) / (Frequency of 12C+ ion) = 14 / 12

Simplifying, we get:

(Frequency of 14C+ ion) / (Frequency of 12C+ ion) = 7 / 6

Therefore, the ratio of the frequencies of revolution for 14C+ ions to 12C+ ions is 7/6.

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