Question

What is the mass percentage of the gold in the alloy? What is the mass percentage of the palladium in the alloy?

A 67.2 g sample of a gold and palladium alloy contains 3.61×1023 atoms .
What is the mass percentage of the gold in the alloy?
What is the mass percentage of the palladium in the alloy?

Answer 1

`11.1%` gold and `88.7%` palladium.

Explanation:

First, we need to find the number of moles of gold and palladium in the alloy.

To do so, we can represent our situation with a system of equations!

Let `x` represent the number of moles of gold in the sample, and let `y` represent the number of moles of palladium.

For our first equation, we know that `"number of atoms = number of moles ⋅ "6.022*10^23"`
So, the number of gold atoms is `x*6.022*10^23` and the number of palladium atoms is `y*6.022*10^23`.
The total number of atoms is given: `3.61*10^23`. Therefore,

`x*6.022*10^23+y*6.022*10^23=3.61*10^23`

For our second equation, we need to understand that the mass of the alloy is equal to the mass of gold added to the mass of palladium.
The mass of gold is `"moles of gold ⋅ mass of 1 mole of gold"`, which is the same thing as `x * "196.97g"`.
The mass of palladium is `"moles of palladium ⋅ mass of 1 mole of palladium"`, which is the same thing as `y * "106.42g"`.

`196.97x + 106.42y = "67.2g"`

So, we just need to solve for `x` and `y` in these `2` equations:

`x*6.022*10^23+y*6.022*10^23=3.61*10^23`
`196.97x + 106.42y = "67.2g"`

1. Isolate `x`.
   `x*6.022*10^23 = 3.61*10^23 - y*6.022*10^23`
   `x = (3.61*10^23 - y*6.022*10^23)/(6.022*10^23)`

2. Substitute `x` in the second equation to solve for `y`.
   `196.97x + 106.42y = "67.2g"`
   `196.97((3.61*10^23 - y*6.022*10^23)/(6.022*10^23)) + 106.42y = 67.2`
   `196.97((3.61*10^23)/(6.022*10^23) - (y*cancel(6.022*10^23))/cancel(6.022*10^23)) + 106.42y = 67.2`
   `196.97(0.599 - y) + 106.42y = 67.2`
   `118.07-196.97y+106.42y= 67.2`
   `118.07-90.55y= 67.2`
   `90.55y=118.07-67.2`
   `y=0.56`

3. Plug in `y` to solve for `x`.
   `196.97x + 106.42y = 67.2`
   `196.97x + 106.42(0.56) = 67.2`
   `196.97x + 59.595 = 67.2`
   `196.97x = 67.2- 59.595`
   `x = 0.038`

Finally, we can find the mass percent of gold and palladium! :)

`"mass percent" = "mass of element"/"mass of sample"`

For gold, `(196.97x)/67.2 = (196.97*0.038)/67.2 = 0.111 = 11.1%`

For palladium, `(106.42y)/67.2 = (106.42*0.56)/67.2 = 0.887 = 88.7%`