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Chapter 5: Problem 58

A quantity of \(0.225 \mathrm{~g}\) of a metal \(\mathrm{M}\) (molar mass = \(27.0 \mathrm{~g} / \mathrm{mol}\) ) liberated \(0.303 \mathrm{~L}\) of molecular hydrogen (measured at \(17^{\circ} \mathrm{C}\) and \(741 \mathrm{mmHg}\) ) from an excess of hydrochloric acid. Deduce from these data the corresponding equation and write formulas for the oxide and sulfate of M.

Short Answer

Expert verified
The balanced equation is \(2M + 6HCl \rightarrow 3H2 + 2MCl3\). The formula for the oxide of M is \(M2O3\) and for the sulfate of M is \(M2(SO4)3\).

Step by step solution

01

Calculate Number of Moles of Hydrogen (H2)

Convert volume of H2 to moles using Avogadro's law, which states that one mole of any gas occupies a volume of \(22.4 L\) at STP. The STP conditions are \(0^{\circ}C\) (which is \(273.15 K\)) and \(1 atm\), but in this case we have different conditions; \(17^{\circ}C\) (which is \(290.15 K\)) and \(741 mmHg\) (which is approximately \(0.97 atm\)). The equation to use is \(PV=nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant and \(T\) is the temperature in Kelvin. Rearranging for n gives \(n = PV/RT\). After substituting given values, we find number of moles: \(n_{H2} = (0.97*0.303)/(0.0821*290.15) = 0.012 moles.\)
02

Calculate Number of Moles of Metal M

Now, calculate the number of moles of metal \(M\). The number of moles (\(n\)) is given by mass divided by molar mass. Using the given mass and molar mass: \(n_M = 0.225/27 = 0.00833 moles\)
03

Deduce the Stoichiometry of the Reaction

Comparing the moles of \(M\) to the moles of \(H2\), one has a ratio of 0.00833 to 0.012, which simplifies to 2:3. This means two moles of \(M\) react to produce three moles of \(H2\). This information is used to write the balanced equation: \(2M + 6HCl \rightarrow 3H2 + 2MCl3\)
04

Write Formulas for the Oxide and Sulfate of M

Considering that \(M\) forms a chloride \(MCl3\) as per the reaction, this means the metal is trivalent, i.e., \(M^{3+}\). Thus, the formula for the oxide of \(M\) must be \(M2O3\) (since the oxide ion \(O^{2-}\) requires three \(M^{3+}\) ions for charge balance), and for the sulfate must be \(M2(SO4)3\) (since the sulfate ion \(SO4^{2-}\) requires three \(M^{3+}\) ions for charge balance).

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