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The element nitrogen has an atomic weight of \( \mathbf{1 4 . 0} \) and consists of two stable isotopes nitrogen-14 and nitrogen-15. The isotope nitrogen-14 has a mass of 14.0 amu and a percent natural abundance of \( 99.6 \% \). The isotope nitrogen-15 has a percent natural abundance of \( 0.370 \% \). What is the mass of nitrogen-15?

Ask by Park Gray. in South Africa
Feb 25,2025

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The mass of nitrogen-15 is approximately 15.135 amu.

Solución

To find the mass of nitrogen-15, we can use the given information about the atomic weight of nitrogen and the percent natural abundance of the two stable isotopes. Given: - Atomic weight of nitrogen = 14.0 amu - Nitrogen-14 has a mass of 14.0 amu and a percent natural abundance of 99.6% - Nitrogen-15 has a percent natural abundance of 0.370% Let's denote the mass of nitrogen-15 as \( m_{15} \). The atomic weight of nitrogen is the weighted average of the masses of the two isotopes based on their natural abundance. We can set up the following equation: \[ 14.0 = (0.996 \times 14.0) + (0.0037 \times m_{15}) \] Now, we can solve for \( m_{15} \) to find the mass of nitrogen-15. Solve the equation by following steps: - step0: Solve for \(m_{15}\): \(14=\left(0.996\times 14\right)+0.0037m_{15}\) - step1: Multiply the numbers: \(14=13.944+0.0037m_{15}\) - step2: Swap the sides: \(13.944+0.0037m_{15}=14\) - step3: Move the constant to the right side: \(0.0037m_{15}=14-13.944\) - step4: Subtract the numbers: \(0.0037m_{15}=0.056\) - step5: Divide both sides: \(\frac{0.0037m_{15}}{0.0037}=\frac{0.056}{0.0037}\) - step6: Divide the numbers: \(m_{15}=\frac{560}{37}\) The mass of nitrogen-15 is approximately 15.135 amu.

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The Deep Dive

To find the mass of nitrogen-15 given its natural abundance, we can use the concept of weighted average based on the isotopes and their respective natural abundances. We know the average atomic weight (14.0 amu) is calculated using the formula: \[ \text{Average atomic weight} = ( \text{mass of N-14} \times \text{abundance of N-14} ) + ( \text{mass of N-15} \times \text{abundance of N-15} ) \] We can plug in our known values: - Mass of N-14 = 14.0 amu - Percent abundance of N-14 = 99.6% = 0.996 - Percent abundance of N-15 = 0.370% = 0.00370 Substituting into the equation gives: \[ 14.0 = (14.0 \times 0.996) + (m \times 0.00370) \] This simplifies to: \[ 14.0 = 13.944 + 0.00370m \] Now, subtract 13.944 from both sides: \[ 0.056 = 0.00370m \] Finally, solve for \(m\): \[ m = \frac{0.056}{0.00370} \approx 15.1351 \, \text{amu} \] Thus, the mass of nitrogen-15 is approximately \( 15.1351 \, \text{amu} \).

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