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Chapter 21: Problem 12

In the operating room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients undergoing surgery. One gas that is often monitored is the anesthetic isoflurane (molecular mass \(\left.=3.06 \times 10^{-25} \mathrm{~kg}\right)\). In a spectrometer, a singly ionized molecule of isoflurane (charge \(=+e\) ) moves at a speed of \(7.2 \times 10^{3} \mathrm{~m} / \mathrm{s}\) on a circular path that has a radius of \(0.10 \mathrm{~m}\). What is the magnitude of the magnetic field that the spectrometer uses?

Short Answer

Expert verified
The magnitude of the magnetic field is approximately 0.138 T.

Step by step solution

01

Identify the Known Values

The mass of the isoflurane molecule is given as \( m = 3.06 \times 10^{-25} \text{ kg} \). The charge of the ionized isoflurane molecule is \( q = e = 1.60 \times 10^{-19} \text{ C} \). The speed of the molecule is \( v = 7.2 \times 10^3 \text{ m/s} \). The radius of its circular path is \( r = 0.10 \text{ m} \).
02

Use the Formula for Magnetic Force

For a charged particle moving in a magnetic field on a circular path, the magnetic force provides the required centripetal force to maintain the circular motion. The relationship is \( F_{magnetic} = qvB = \frac{mv^2}{r} \), where \( B \) is the magnetic field strength.
03

Rearrange the Formula

Re-arrange the formula to solve for the magnetic field \( B \): \( B = \frac{mv}{qr} \).
04

Substitute the Known Values

Substitute the known values into the equation: \( B = \frac{(3.06 \times 10^{-25})(7.2 \times 10^3)}{(1.60 \times 10^{-19})(0.10)} \).
05

Perform the Calculation

Carry out the calculation: \( B = \frac{3.06 \times 10^{-25} \times 7.2 \times 10^3}{1.60 \times 10^{-19} \times 0.10} = \frac{2.2032 \times 10^{-21}}{1.60 \times 10^{-20}} = 0.1377 \text{ T} \).
06

Round and State the Result

The magnitude of the magnetic field used by the spectrometer is approximately \( 0.138 \text{ T} \) when rounded to three significant figures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Anesthetic Gases
Anesthetic gases are crucial in the field of medicine, particularly during surgical procedures. These gases serve the primary purpose of inducing unconsciousness and preventing pain, allowing surgeries to be conducted without the patient experiencing discomfort. Isoflurane is one such anesthetic gas frequently used in operating rooms. This inhaled agent works by affecting the central nervous system, ensuring that patients remain in a controlled restful state throughout the procedure.

Mass spectrometry is an essential tool for anesthesia monitoring. It provides accurate measurement of the concentration of gases administered to patients. This real-time monitoring ensures that the correct dosage of anesthetic is given. Mistakes in dosage can lead to complications, so precision is critical.
  • Helps in adjusting doses quickly and accurately.
  • Improves patient safety by preventing overdose.
  • Ensures optimum gas balance is maintained during surgery.
Understanding the role of mass spectrometers in anesthesia highlights the importance of technology in enhancing patient care.
Magnetic Field Calculation
The process of calculating a magnetic field in a mass spectrometer is crucial for its operation. When a charged particle, such as an ionized molecule, moves through a magnetic field, it experiences a force that causes it to travel in a circular path. This is the principle that allows a mass spectrometer to function, as the radius of this path can be used to determine the properties of the particle.

In the exercise, you're given the formula for the magnetic force which is: \[F_{magnetic} = qvB = \frac{mv^2}{r}\]
This formula is re-arranged to solve for the magnetic field strength \( B \): \[B = \frac{mv}{qr}\]
By substituting known values:
  • Mass \( m = 3.06 \times 10^{-25} \, \mathrm{kg} \)
  • Charge \( q = 1.60 \times 10^{-19} \, \mathrm{C} \)
  • Velocity \( v = 7.2 \times 10^3 \, \mathrm{m/s} \)
  • Radius \( r = 0.10 \, \mathrm{m} \)
This yields the magnetic field \( B = 0.138 \, \mathrm{T} \).
Understanding these calculations allows for accurate control of the device and precision in measuring various substances.
Ionized Molecules
In the context of mass spectrometry, ionized molecules are central to the functioning of the device. Ionization refers to the process where a molecule gains or loses electrons, resulting in a charged species known as an ion. This charge is vital as it allows the molecule to be manipulated by electric and magnetic fields within the spectrometer.

For example, the exercise gives us an isoflurane molecule that is singly ionized, meaning it has gained or lost one electron and carries a charge of \(+e\). This charge interacts with the magnetic field, causing the molecule to deflect from its original path onto a circular trajectory.

The behavior of ionized molecules is critical in identifying unknown substances. Mass spectrometers differentiate ions based on their mass-to-charge ratio, allowing for detailed analysis of molecular structures. The lightest ions are deflected the most, while heavier ions travel in larger circuits within the device.
  • Ionization enables separation based on mass.
  • Allows for precise determination of molecular weights.
  • Facilitates analysis of complex mixtures.
Through the understanding of ionization, scientists can decode the composition of substances with high precision and accuracy.

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Most popular questions from this chapter

The \(x, y,\) and \(z\) components of a magnetic field are \(B_{x}=0.10 \mathrm{~T}, B_{y}=0.15 \mathrm{~T},\) and \(B_{z}=0.17 \mathrm{~T}\). A \(25-\mathrm{cm}\) wire is oriented along the \(z\) axis and carries a current of \(4.3 \mathrm{~A}\) What is the magnitude of the magnetic force that acts on this wire?

A proton is projected perpendicularly into a magnetic field that has a magnitude of \(0.50 \mathrm{~T}\). The field is then adjusted so that an electron will follow the exact same circular path when it is projected perpendicularly into the field with the same velocity that the proton had. What is the magnitude of the field used for the electron? Verify that your answer is consistent with your answers to the Concept Questions.

Two insulated wires, each \(2.40 \mathrm{~m}\) long, are taped together to form a two-wire unit that is \(2.40 \mathrm{~m}\) long. One wire carries a current of \(7.00 \mathrm{~A} ;\) the other carries a smaller current \(I\) in the opposite direction. The two-wire unit is placed at an angle of \(65.0^{\circ}\) relative to a magnetic field whose magnitude is \(0.360 \mathrm{~T}\). The magnitude of the net magnetic force experienced by the two-wire unit is \(3.13 \mathrm{~N}\). What is the current \(I ?\)

The drawing shows an end-on view of three wires. They are long, straight, and perpendicular to the plane of the paper. Their cross sections lie at the corners of a square. The currents in wires 1 and 2 are \(I_{1}=I_{2}=I\) and are directed into the paper. What is the direction of the current in wire \(3,\) and what is the ratio \(I_{3} / I,\) such that the net magnetic field at the empty corner is zero?

Consult Interactive Solution \(\underline{21.43} 21.43\) at to see how this problem can be solved. The coil in Figure \(21-22 a\) contains 410 turns and has an area per turn of \(3.1 \times 10^{-3} \mathrm{~m}^{2}\). The magnetic field is \(0.23 \mathrm{~T},\) and the current in the coil is \(0.26 \mathrm{~A} .\) A brake shoe is pressed perpendicularly against the shaft to keep the coil from turning. The coefficient of static friction between the shaft and the brake shoe is \(0.76 .\) The radius of the shaft is \(0.012 \mathrm{~m}\). What is the magnitude of the minimum normal force that the brake shoe exerts on the shaft?
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