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Please help me answering this question. i will rate the every correct answer

Elements that appear in the same column of the periodic table often share similar chemical properties. In the case of the alkaline earth metals, this is troublesome since the body treats calcium (necessary for proper bone growth) and radium (a radioactive element) as chemically similar, storing both in bone marrow. The radium then bombards nearby bone cells with alpha particles, causing them to "crumble." Radium poisoning investigations often center on the identification of radium and its isotopes in bone samples using a mass spectrometer. Pictured is a schematic of a simplified mass spectrometer, showing the paths of calcium, barium (another alkaline earth metal) and radium isotopes entering the chamber. The region shown is immersed in a constant magnetic field of 0.352 tesla pointing out of the plane of the schematic. Motion of the positively-charged isotopes toward the right was initiated by a charge separation of 2876 volts on the two plates shown. Using the data shown in the table below, calculate the path radius of the Ca+ ion. Note that 1 amu = 1.661 times 10^-27 kg and 1 e = 1.602 times 10^-19 C. Using the same data table, match the particles to their path label. Path A Path B Path C Path D Path E Path F Path G

Explanation / Answer

Path radius of Ca+ ion:

m = 40.1 amu = 40.1 x 1.661x10-27 Kg = 6.66x10-26 Kg

q = +1e = 1.602x10-19 C

voltage, V = 2876 V

velocity (v) of the charged particle can be found from the law of conservation of energy as

(1/2)mxv2 = q x V

=> v = (2qV / m)1/2 =

=> v = (2x1.602x10-19 Cx2876 V / 6.66x10-26 Kg)1/2

=> v = 1.176x105 m/s

Now we can calculate the radius from the following formulae

r = mv / qB

=> r = 6.66x10-26 Kg x1.176x105 m.s-1 / (1.602x10-19 Cx0.352 T)

=> r = 0.139 m = 13.9 cm (answer)

Q.2:

r = mv / qB

=> r = m x (2qV / m)1/2 / qB

=> r = (2mV / qB2)1/2

Since V and B are constant, 'r' is proportional to (m / q)1/2  

Hence higher the (m/q) value, higher is the radius

For Ca+, m/q = 40.1 amu / 1 e = 40.1 amu/e

For Ca2+, m/q = 40.1 amu / 2 e = 20.05 amu/e

For Ba+, m/q = 137 amu / 1 e = 137 amu/e

For Ba2+, m/q = 137 amu / 2 e = 68.5 amu/e

For Ra+, m/q = 226 amu / 1 e = 226 amu/e

For Ra2+, m/q = 226 amu / 2 e = 113 amu/e

For Ra3+, m/q = 226 amu / 3 e = 75.33 amu/e

Hence

Path - A (smallest radius) --- > Ca2+

Path - B ---- > Ca+

Path - C ---- > Ba2+

Path - D ---- > Ra3+

Path - E ----- > Ra2+

Path - F ------ > Ba+

Path - G ----- > Ra+

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