# Finding atomic weight from vapor density of metal chloride

## Homework Statement

Vapor density of a metal chloride is 66. Its oxide contains 53% metal. The atomic weight of the metal is ?

## Homework Equations

vapor density = 0.5 * molar mass

## The Attempt at a Solution

Using the above formula, I found the molar mass of the metal chloride as 132. I am stuck here, as the problem talks about the oxide containing 53% metal.

In 100 g of Metal oxide, 47% is oxygen, i.e. roughly the weight of three oxygen atoms which is 48. The Formula is therefore M2O3 ? Does this mean that this metal is hexa valent? This doesn't seem right, because the mol wt of chlorine is 35.5 and six chlorine atoms would weigh 213 g whereas the sample is only 132 g.

The choices for answer are 21, 54, 27.06 and 2.7.

Logically I think the answer should be 54, as this would correspond to cobalt while the rest are not metals.

Can anybody please explain how the answer is to be arrived at?

density is a physical quantity with units. you had presented a unitless quantity for it.

Vapor density is a ratio, being a proportion of mass. Therefore no unit.

Now, how do I go about solving the problem?

Vapor density is a ratio, being a proportion of mass.

Proportion between what masses?

Vapour density (or vapor density) is the density of a vapour in relation to that of hydrogen. It may be defined as mass of a certain volume of a substance divided by mass of same volume of hydrogen.

vapour density = mass of n molecules of gas / mass of n molecules of hydrogen

(By definition, the molar mass of a gas is the ratio of the mass of one molecule of gas to that of an hydrogen atom under similar conditions.)

Therefore:

vapour density = molar mass of gas / molar mass of H2

vapour density = molar mass of gas / 2

vapour density = ½ × molar mass

(and thus: molar mass = 2 × vapour density)

Source: http://en.wikipedia.org/wiki/Vapour_density" [Broken]

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Is the content of metal in the oxide given by mass or by number?

I have reproduced the question exactly. It has to be 53% by mass since molar mass is involved.

Sorry if I sound rude, but the discussion doesn't seem to be going anywhere.

Suppose the valency of the metal is v and its atomic mass is x. Then the ratio of the masses of the metal and oxygen in the oxide is:

$$\frac{2 \, x}{16 \times v} = \frac{53}{47} \Rightarrow 94 \, x = 848 \, v \Rightarrow 47 \, x = 424 \, v$$

Next, the molar mass of the chloride is:
$$1 \times x + v \times 35 = 66 \Rightarrow x + 35 \, v = 66$$
where I had used the definition of vapour density.

Now, you have 2 linear equations in 2 unknowns (x and v). Solve them

or, if the vapour density is defined with respect to H2, then you should substitute the 66 in the second equation with a 132.

Suppose the valency of the metal is v and its atomic mass is x. Then the ratio of the masses of the metal and oxygen in the oxide is:

Thanks for the reply, but I still don't understand. The ratio of the masses of the metal and the oxide is 53:47 as per the question. How do I relate it to valency and atomic mass and form an equation with v and x?

Next, the molar mass of the chloride is:
1 \times x + v \times 35 = 66 \Rightarrow x + 35 \, v = 66
where I had used the definition of vapour density.

Now, you have 2 linear equations in 2 unknowns (x and v). Solve them

Can you make it a bit clearer?

No, this is enough hints. If you cannot solve it, that's your problem.