2. vanadium has two naturally occurring isotopes, $^{50}v$ with an atomic mass of 49.9472 amu and $^{51}v$…

2. vanadium has two naturally occurring isotopes, $^{50}v$ with an atomic mass of 49.9472 amu and $^{51}v$ with an atomic mass of 50.9440 amu. the atomic mass of vanadium is 50.9415 amu. the percent abundances of the vanadium isotopes are ______% $^{50}v$ and ______% $^{51}v$.\na) 49, 51\nb) 99.75, 0.25\nc) 99, 1.0\nd) 1.0, 99\ne) 0.25, 99.75

2. vanadium has two naturally occurring isotopes, $^{50}v$ with an atomic mass of 49.9472 amu and $^{51}v$ with an atomic mass of 50.9440 amu. the atomic mass of vanadium is 50.9415 amu. the percent abundances of the vanadium isotopes are ______% $^{50}v$ and ______% $^{51}v$.\na) 49, 51\nb) 99.75, 0.25\nc) 99, 1.0\nd) 1.0, 99\ne) 0.25, 99.75

Answer

Explanation:

Step1: Let the percent abundance of $^{50}V$ be $x$, then the percent abundance of $^{51}V$ is $1 - x$.

Let the percent - abundance of $^{50}V$ be $x$ (in decimal form), so the percent - abundance of $^{51}V$ is $1 - x$. The average atomic mass formula is $\text{Average atomic mass}=\sum_{i}m_{i}x_{i}$, where $m_{i}$ is the atomic mass of the isotope and $x_{i}$ is its abundance.

Step2: Set up the equation using the average atomic mass formula.

We know that the atomic mass of $^{50}V$ is $m_1 = 49.9472$ amu, the atomic mass of $^{51}V$ is $m_2=50.9440$ amu, and the average atomic mass of vanadium is $50.9415$ amu. So the equation is $49.9472x + 50.9440(1 - x)=50.9415$.

Step3: Expand and simplify the equation.

Expand the left - hand side: $49.9472x+50.9440 - 50.9440x = 50.9415$. Combine like terms: $(49.9472x-50.9440x)+50.9440 = 50.9415$, which gives $- 0.9968x+50.9440 = 50.9415$.

Step4: Solve for $x$.

Subtract $50.9440$ from both sides: $-0.9968x=50.9415 - 50.9440=-0.0025$. Then $x=\frac{-0.0025}{-0.9968}\approx0.0025$.

Step5: Convert $x$ to a percentage and find the percentage of $^{51}V$.

The percentage of $^{50}V$ is $x\times100 = 0.25%$. The percentage of $^{51}V$ is $(1 - x)\times100=(1 - 0.0025)\times100 = 99.75%$.

Answer:

E. 0.25, 99.75