Finding atomic weight from vapor density of metal chloride

[Quantum Mind]

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?

 

[Dickfore]

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

 

[Quantum Mind]

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

Now, how do I go about solving the problem?

 

[Dickfore]

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

Proportion between what masses?

 

[Quantum Mind]

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"

 

[Dickfore]

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

 

[Quantum Mind]

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.

 

[Dickfore]

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

 

[Dickfore]

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

 

[Quantum Mind]

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?

 

[Dickfore]

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