How do I find the ratio of the reaction of the residue and HCl?

A mixture weighing $4.08\ \mathrm g$ of $\ce{BaO}$ ($M(\ce{Ba})$ = $138\ \mathrm{g\ mol^{-1}}$) and an unknown carbonate $\ce{XCO3}$ was heated strongly. The residue weighed $3.64\ \mathrm g$. This was dissolved in $100\ \mathrm{ml}$ of $1\ \mathrm M$ $\ce{HCl}$. The excess acid required $16\ \mathrm{ml}$ of $2.5\ \mathrm M$ $\ce{NaOH}$ solution for complete neutralization. What is the molar mass of $\ce{X}?$

My approach:

$16\ \mathrm{ml}$ of $2.5\ \mathrm M$ $\ce{NaOH}$ means $0.04\ \mathrm{mol}$ of $\ce{NaOH}$. Thus, the excess acid was $0.04\ \mathrm{mol}$ of $\ce{HCl}$. $100\ \mathrm{ml}$ of $1\ \mathrm M$ $\ce{HCl}$ means $0.1\ \mathrm{mol}$ of $\ce{HCl}$. Thus, $0.06\ \mathrm{mol}$ of $\ce{HCl}$ reacted with the residue. Now, how can I find the ratio in which the residue reacted with $\ce{HCl}?$ What is the composition of the residue?

• The part you've already managed to do looks good. Now, notice that you can create a system of three equations (initial mass balance / mass balance after thermal decomposition / acid reaction stoichiometry) in three variables (the mole amount of $\ce{BaO}$ / the mole amount of unknown carbonate / the mass of unknown carbonate) with the remaining data. – Nicolau Saker Neto Nov 23 '13 at 15:51
• I am unable to follow. Could you give one such equation? – Tejas Nov 23 '13 at 16:00
• I'll give the middle equation. $\ce{BaO}$ is not altered with heating, so the same initial mass $m_{BaO}$ is present. The carbonate decomposes on heating according to $\ce{XCO3 → XO + CO2}$. Therefore, an initial mass $m_{XCO_3}$ composed of $n_{XCO_3}$ mols becomes $n_{XO}=n_{XCO_3}$ mols of $\ce{XO}$ with mass $m_{XO}$. We know that the relation between the molar masses of $\ce{XO}$ and $\ce{XCO3}$ is $M_{XO}=M_{XCO_3}-M_{CO_2}=M_{XCO_3}-44\ g/mol$. – Nicolau Saker Neto Nov 23 '13 at 18:28
• With this information, we can convert $m_{BaO}+m_{XO}=3.64\ g$ to a function of $m_{BaO}$, $m_{XCO_3}$ and $n_{XCO_3}$ by using $m_{XO}=M_{XO}\times n_{XO}=(M_{XCO_3}-44\ g/mol)\times n_{XCO_3}=m_{XCO_3}-44n_{XCO_3}\ g/mol$. Hence, a possible form for the second equation is $m_{BaO}+m_{XCO_3}-44n_{XCO_3}\ g/mol=3.64\ g$. You can rewrite the equation if you choose to work mostly with masses or with mole amounts, it's up to you how to write and solve it. – Nicolau Saker Neto Nov 23 '13 at 18:28

In the comments I mentioned one way to solve the problem, but it turns out that was the roundabout way! There is a simpler solution, as follows.

From the neutralization reaction, it is possible to calculate that the mixture reacted with $0.06\ \mathrm{mol}$ of $\ce{H+}$. The carbonate and the oxide react in a proportion of 1:2 with the hydrogen ions, so we know that $n_{\ce{BaO}}+n_{\ce{XO}}=0.03\ \mathrm{mol}$. The mass balance from the thermal decomposition then suggests that $m_{\ce{XCO3}}-m_{\ce{XO}}=0.44\ \mathrm g$. These two equations are already enough.

Notice that the for the molar masses of the unknown oxide and carbonate, we have $M_{\ce{XCO3}}-M_{\ce{XO}}=44\ \mathrm{g/mol}$. If one converts the masses in the second equation into expressions in $M$ and $n$, the result is $n_{\ce{XCO3}}M_{\ce{XCO3}}-n_{\ce{XO}}M_{\ce{XO}}=0.44\ \mathrm g$, which can be turned into $n_{\ce{XCO3}}M_{\ce{XCO3}}-n_{\ce{XO}}\left(M_{\ce{XCO3}}-44\ \mathrm{g/mol}\right)=0.44\ \mathrm g$. It is easy to see that the thermal decomposition does not alter the number of moles of compound containing $\ce{X}$, so that $n_{\ce{XCO3}}=n_{\ce{XO}}$. Thus:

$$n_{\ce{XCO3}}M_{\ce{XCO3}}-n_{\ce{XCO3}}\left(M_{\ce{XCO3}}-44\ \mathrm{g/mol}\right)=0.44\ \mathrm g$$

$$n_{\ce{XCO3}}\left(M_{\ce{XCO3}}-M_{\ce{XCO3}}+44\ \mathrm{g/mol}\right)=0.44\ \mathrm g$$

$$n_{\ce{XCO3}}\times 44\ \mathrm{g/mol}=0.44\ \mathrm g\quad\Rightarrow\quad n_{\ce{XCO3}}=0.01\ \mathrm{mol}$$

Now the problem unravels easily. We get that $n_{\ce{BaO}}=0.02\ \mathrm{mol}$, which implies $m_{\ce{BaO}}=3.08\ \mathrm g$ and $m_{\ce{XCO3}}=1.00\ \mathrm g$. Hence, $M_{\ce{XCO3}}=100\ \mathrm{g/mol}$, and from there $M_{\ce{X}}=40\ \mathrm{g/mol}$. A quick look on a periodic table yields calcium as the most likely candidate, and indeed its carbonate has the formula $\ce{CaCO3}$ and a molar mass of $100\ \mathrm{g/mol}$.