Question

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg}$$ A sample containing sulfur (atomic mass 32 u), manganese \((55 \mathrm{u}),\) and an unknown element is placed in a mass spectrometer. The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field. The sulfur and manganese lines are separated by \(3.20 \mathrm{cm},\) and the unknown element makes a line between them that is \(1.07 \mathrm{cm}\) from the sulfur line. (a) What is the mass of the unknown clement? (b) Identify the element.

Short answer

Answer: The mass of the unknown element is approximately 48.14 u, and the element is cadmium (Cd).

Step-by-step solution

Understanding the principle of the mass spectrometer

A mass spectrometer works on the principle that ions with the same charge but different masses will follow different trajectories when subjected to a magnetic field. In this problem, we assume that all ions have the same charge and are accelerated through the same potential difference, so their velocities entering the magnetic field will be proportional to the square root of their mass.

Analyze the given distances

We are given the distance between the sulfur and manganese lines as 3.20 cm, and the distance from the sulfur line to the unknown element's line as 1.07 cm. Let's denote the distance between the manganese line and the unknown element's line as x. Then we can write an equation relating the three distances: $$x = 3.20 - 1.07$$

Calculate the distance x

Now we calculate the distance x between the manganese line and the unknown element's line: $$x = 3.20 - 1.07$$ $$x = 2.13 \mathrm{cm}$$

Relate the distances to the masses

Since the velocities entering the magnetic field are proportional to the square root of the masses, we can relate the distances to the masses as follows: $$\frac{1.07}{3.20} = \frac{\sqrt{M_S}}{\sqrt{M_{Mn}}} \quad \text{and} \quad \frac{2.13}{3.20} = \frac{\sqrt{M_{U}}}{\sqrt{M_{Mn}}}$$ where \(M_S\) is the mass of sulfur, \(M_{Mn}\) is the mass of manganese, and \(M_U\) is the mass of the unknown element.

Solve for the mass of the unknown element

We can solve for the mass of the unknown element \(M_U\) by using the given atomic masses for sulfur and manganese: $$\frac{1.07}{3.20} = \frac{\sqrt{32}}{\sqrt{55}} \quad \text{and} \quad \frac{2.13}{3.20} = \frac{\sqrt{M_{U}}}{\sqrt{55}}$$ Now we can solve for \(M_U\): $$M_U = 55 \left(\frac{2.13}{3.20}\right)^2$$ $$M_U \approx 48.14 \mathrm{u}$$

Identify the unknown element

With a mass of approximately 48.14 u, the unknown element is most likely cadmium (Cd), which has an atomic mass of 48 u. So the unknown element is cadmium (Cd).