To five significant figures, what are the cyclotron frequencies in a 3.0000 T magnetic field of the ions (a) O₂⁺, (b) N₂⁺, and (c) CO⁺? The atomic masses are shown in the table; the mass of the missing electron is less than 0.001 u and is not relevant at this level of precision. Although N₂⁺ and CO⁺ both have a nominal molecular mass of 28, they are easily distinguished by virtue of their slightly different cyclotron frequencies. Use the following constants: 1 u = 1.6605 x 10⁻²⁷ kg, e = 1.6022 x 10⁻¹⁹ C.

Verified step by step guidance

1. Step 1: Understand the cyclotron frequency formula. The cyclotron frequency (f) is given by the formula: $f=\frac{q}{m}B$, where q is the charge of the ion, m is the mass of the ion, and B is the magnetic field strength.
2. Step 2: Determine the charge of the ions. Since the ions are singly charged (O₂⁺, N₂⁺, CO⁺), the charge q is equal to the elementary charge, $e=1.6022\times {10}^{-}C$.
3. Step 3: Calculate the mass of each ion in kilograms. Use the atomic masses provided in the table and the conversion factor $1u=1.6605\times {10}^{-}\mathrm{kg}$. For O₂⁺, the mass is $2\times 15.995\times 1.6605\times {10}^{-}$. Similarly, calculate for N₂⁺ and CO⁺ using their respective atomic masses.
4. Step 4: Substitute the values into the cyclotron frequency formula. For each ion, substitute q = $1.6022\times {10}^{-}$, m (calculated in Step 3), and B = $3.0000T$ into the formula $f=\frac{q}{m}B$.
5. Step 5: Compare the cyclotron frequencies of N₂⁺ and CO⁺. Note that although both have a nominal molecular mass of 28 u, their slightly different atomic masses will result in distinct cyclotron frequencies, allowing them to be distinguished.

Key Concepts

Cyclotron Frequency

Cyclotron frequency is the frequency at which a charged particle orbits in a magnetic field. It is determined by the charge of the particle, the strength of the magnetic field, and the mass of the particle. The formula for cyclotron frequency (f) is f = (qB)/(2πm), where q is the charge, B is the magnetic field strength, and m is the mass of the particle. This concept is crucial for understanding the motion of ions in magnetic fields.

Atomic Mass Unit (u)

The atomic mass unit (u) is a standard unit of mass used to express atomic and molecular weights. One atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom, approximately 1.6605 x 10⁻²⁷ kg. In this context, the atomic masses of ions like O₂⁺, N₂⁺, and CO⁺ are essential for calculating their cyclotron frequencies, as the mass directly influences the frequency of their motion in a magnetic field.

Magnetic Field Strength

Magnetic field strength, denoted as B, is a measure of the intensity of a magnetic field at a given point in space. It is typically measured in teslas (T). In this problem, a magnetic field strength of 3.0000 T is provided, which is necessary for calculating the cyclotron frequencies of the ions. The strength of the magnetic field affects the radius and frequency of the charged particles' circular motion.