Question

A mass spectrometer is being used to separate common oxygen-16 from the much rarer oxygen-18, taken from a sample of old glacial ice. (The relative abundance of these oxygen isotopes is related to climatic temperature at the time the ice was deposited.) The mass of oxygen-16 is 2.66 X 10^-26 kg and the mass of oxygen-18 is 2.99 X 10^-26 kg. Assuming are ionized such that they are singly charged and travel at 2 X 10⁶ m/s in a 0.8-T magnetic field. What is the acceleration of the two different oxygen isotopes?

Answer 1

Answer 2

Given:

• Velocity ($v$): $2\times 10^6\text{m/s}$

• Magnetic field ($B$): $0.8\text{T}$

• Charge ($q$): $1.6\times 10^{-19}\text{C}$ (singly charged ions)

• Masses:

• Oxygen-16 ($m_{16}$): $2.66\times 10^{-26}\text{kg}$

• Oxygen-18 ($m_{18}$): $2.99\times 10^{-26}\text{kg}$

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Step-by-Step Calculation:

1. Magnetic Force ($F$) on Each Ion:

$$F=qvB=(1.6\times 10^{-19})(2\times 10^6)(0.8)=2.56\times 10^{-13}\text{N}$$

 

2. Acceleration ($a$) for Each Isotope:

• Oxygen-16:

  $$a_{16}=\frac{F}{m_{16}}=\frac{2.56\times 10^{-13}}{2.66\times 10^{-26}}=9.62\times 10^{12}{\text{m/s}}^2$$

• Oxygen-18:

  $$a_{18}=\frac{F}{m_{18}}=\frac{2.56\times 10^{-13}}{2.99\times 10^{-26}}=8.56\times 10^{12}{\text{m/s}}^2$$

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Answer:

• Oxygen-16 acceleration: $9.62\times 10^{12}{\text{m/s}}^2$

• Oxygen-18 acceleration: $8.56\times 10^{12}{\text{m/s}}^2$