An unknown element Q has two known isotopes: 60Q and 63Q. If the average atomic mass is 60.5 amu, what are the relative percentages of the isotopes?
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To find the relative percentages of the isotopes, we can set up a system of two equations using the average atomic mass and the relative abundance of the isotopes.
Let's denote the relative abundance of the isotope 60Q as x, and the relative abundance of the isotope 63Q as (1 - x).
The average atomic mass is given as 60.5 amu, which is calculated as follows:
(60Q * x) + (63Q * (1 - x)) = 60.5
Now, let's solve this equation to find the value of x:
60Q * x + 63Q - 63Q * x = 60.5
60Qx - 63Qx = 60.5 - 63Q
-3Qx = 60.5 - 63Q
-3Qx + 63Q = 60.5
Q(-3x + 63) = 60.5
Q = 60.5 / (-3x + 63)
The relative abundance of 60Q is equal to x, so substituting the value of Q back into the equation:
60Q = 60 * (60.5 / (-3x + 63))
Simplifying further:
60Q = 3630 / (-3x + 63)
60Q = 3630 / (-3(x - 21))
60Q = -1210 / (x - 21)
Now, we can solve for the value of x by setting the equation above equal to the relative abundance of the isotope 60Q, which is x:
x = -1210 / (x - 21)
To solve this equation, we can cross-multiply and simplify:
x(x - 21) = -1210
x^2 - 21x = -1210
x^2 - 21x + 1210 = 0
Now, we can solve this quadratic equation for x. Using factoring or the quadratic formula, we find two possible values for x:
x = 10 or x = 11
Since the relative abundance of an isotope is usually expressed as a percentage, we multiply x by 100 to find the percentages:
Percentage of 60Q = 10%
Percentage of 63Q = 90%
Therefore, the relative percentages of the isotopes 60Q and 63Q are 10% and 90%, respectively.
To find the relative percentages of the isotopes, we need to set up an equation using the average atomic mass and the masses of the isotopes.
Let's assume that x represents the percentage of the 60Q isotope, and therefore (100 - x) represents the percentage of the 63Q isotope.
The average atomic mass is the weighted average of the masses of the isotopes, taking into account their relative abundance:
Average atomic mass = (mass of isotope 1 × percentage of isotope 1 + mass of isotope 2 × percentage of isotope 2) / 100
Given:
Average atomic mass = 60.5 amu
Mass of 60Q isotope = 60 amu
Mass of 63Q isotope = 63 amu
Using the formula, we can set up the following equation:
(60 × x + 63 × (100 - x)) / 100 = 60.5
Simplifying the equation:
(60x + 6300 - 63x) / 100 = 60.5
Combine like terms:
(63 - 60)x = 60.5 × 100 - 63 × 100
(63 - 60)x = 6050 - 6300
-3x = -250
Divide both sides by -3:
x = -250 / -3
x ≈ 83.33
Therefore, the relative percentage of the 60Q isotope is approximately 83.33%, and the relative percentage of the 63Q isotope is approximately 16.67%.
There is little point in posting the same question twice within a couple of minutes.
Let the fraction of 60Q be x and therefore the fraction of 63Q is 1-x.
60x +(1-x)63=60.5
can you take it from here?