Question

Explain how a mass spectrometer, capable of distinguishing between ions with \(\mathrm{m} / \mathrm{e}\) values differing by 1 part in 50,000 , could be used to tell whether an ion of mass 29 is \(\mathrm{C}_{2} \mathrm{H}_{5} \oplus\) or \(\mathrm{CHO}^{\oplus}\)

Short answer

The mass-to-charge ratios of \(C_2H_5^+\) and \(CHO^+\) ions are 29.048 and 29.018, respectively. The mass difference between them is 0.03. Given the mass spectrometer's resolution of 1 part in 50,000, its minimum mass difference that it can resolve is about \(5.81 \times 10^{-4}\), which is smaller than the mass difference between the two ions. Therefore, the mass spectrometer can distinguish between the \(C_2H_5^+\) and \(CHO^+\) ions.

Step-by-step solution

Calculate the mass-to-charge ratios of the ions

To determine whether the mass spectrometer can differentiate these ions, first, we need to find their respective mass-to-charge ratios. Here are the atomic masses of each element: - Carbon (C): 12.01 u - Hydrogen (H): 1.008 u - Oxygen (O): 16.00 u - Charge (e): 1 (since both ions have a charge of +1) Now, we can calculate the mass-to-charge ratio for each ion: - \( C_2H_5^+ \): mass = (2)(12.01) + (5)(1.008) = 29.048 u; m/e = 29.048 - \( CHO^+ \): mass = 12.01 + 1.008 + 16.00 = 29.018 u; m/e = 29.018

Calculate the mass difference between the ions

Next, we need to find the mass difference between the two ions. To do this, we will take the absolute difference between their m/e values: \( \Delta m = |m_{C_2H_5^+} - m_{CHO^+}| = |29.048 - 29.018| = 0.03 \)

Determine if the mass spectrometer can resolve the ions

We are given that the mass spectrometer can distinguish between ions with m/e values that differ by 1 part in 50,000. The minimum mass difference the instrument can resolve can be found by dividing the ion's mass by 50,000: \( \Delta m_{min} = \frac{m_{C_2H_5^+}}{50,000} = \frac{29.048}{50,000} \approx 5.81 \times 10^{-4} \)

Compare the calculated mass difference with the instrument's resolving power

Now, we need to compare the mass difference we calculated in Step 2 with the instrument's resolving power calculated in Step 3: \( 0.03 > 5.81 \times 10^{-4} \) Since the mass difference between the two ions (0.03) is much greater than the minimum mass difference the mass spectrometer can resolve (5.81 x 10^-4), the instrument is capable of distinguishing between the \( C_2H_5^+ \) and \( CHO^+ \) ions.

Key concepts

Mass-to-Charge Ratio

In mass spectrometry, the mass-to-charge ratio, often represented as \( m/e \), is a crucial concept. It signifies the ratio of the ion's mass to its charge, typically measured in atomic mass units (u) per charge (e). Every ion is assigned a \( m/e \) value based on its respective mass and charge.

The mass-to-charge ratio is fundamental because it helps in identifying different ions. For the ions \( C_2H_5^+ \) and \( CHO^+ \), the \( m/e \) values were determined using the atomic masses of the constituent elements.
- \( C_2H_5^+ \): Composed of two carbon atoms and five hydrogen atoms, leading to a mass of 29.048 u.
- \( CHO^+ \): Consists of one carbon, one hydrogen, and one oxygen atom, resulting in a mass of 29.018 u.

While both ions have the same charge of +1, their \( m/e \) ratios slightly differ, mainly noticed in high-resolution conditions.

Ion Differentiation

Ion differentiation in mass spectrometry is about distinguishing ions based on their distinct \( m/e \) values. This process allows scientists to identify and analyze substances in a mixture.

Ions like \( C_2H_5^+ \) and \( CHO^+ \), despite having similar masses, can be differentiated due to their different structural compositions, impacting their \( m/e \) values. The mass difference between them, 0.03 units, might seem negligible; however, it is significant when processed through high-resolution instruments.

• This differentiation is critical in fields like chemistry and biochemistry, where precise identification of molecular structures is necessary.
• It's also essential in forensics, pharmaceuticals, and environmental science for accurate compound analysis.

Instrument Resolving Power

The resolving power of a mass spectrometer defines its capability to differentiate between ions with closely spaced \( m/e \) ratios. A high resolving power indicates that the instrument can distinguish between ions that are very close in mass, crucial for accurately identifying substances.

In the given problem, the resolving power is tested as the instrument can detect ions differing by 1 part in 50,000. Calculating this, we used the formula \( \Delta m_{min} = \frac{m}{50,000} \), where \( m \) is the ion's mass. For the \( C_2H_5^+ \) ion, it was essential to determine if the spectrometer’s resolving power of 5.81 x 10\(^{-4} \) could resolve the 0.03 mass difference with \( CHO^+ \).
This high resolving power is what makes advanced mass spectrometers exceedingly useful in research and industrial applications.

Atomic Mass Calculation

Calculating the atomic mass of a molecule, such as ions in mass spectrometry, involves adding up the atomic masses of their constituent elements. This calculation is vital for identifying the \( m/e \) ratio.

For example, to find the atomic mass of \( C_2H_5^+ \), the masses of two carbon atoms and five hydrogen atoms are summed, leading to a result of 29.048 u. Similarly, \( CHO^+ \) requires the atomic masses of carbon, hydrogen, and oxygen to be combined, resulting in a mass of 29.018 u.
- Atomic mass calculations require knowledge of atomic mass units (u) for each element.
- The process underscores the importance of precision as small errors can lead to incorrect differentiation during ion analysis.

Understanding the calculation of atomic mass is key in mass spectrometry where distinguishing between nearly identical \( m/e \) values is often necessary.