Problem 4

Determining atomic masses. (4.1) The reaction of element X with hydrogen gives compounds analogous to hydrocarbons. 5.000 g of X form 5.628 g of a molar 2:1 mixture of the stoichiometric X-analogues of methane and ethane. Determine the molar mass of X, give its symbol and the 3D structures of the two products. (4.2) The mineral argyrodite is a stoichiometric compound containing silver (oxidation state +1), sulfur (−2) and an unknown element Y (+4). The mass ratio is

m(\ce{Ag}):m(\ce{Y}) = 11.88:1

. Y forms a reddish-brown lower sulfide (Y in +2) and a white higher sulfide (Y in +4). When argyrodite is heated in a stream of hydrogen, the coloured lower sulfide sublimes; the residues are

\ce{Ag2S}

and

\ce{H2S}

. Complete conversion of 10.0 g of argyrodite needs 0.295 dm

^3

\ce{H2}

at 400 K and 100 kPa. Determine the molar mass of Y, its symbol, and the empirical formula of argyrodite. (4.3) IR frequencies (wavenumbers) follow Hooke's law

\tilde{\nu} = \frac{1}{2\pi c}\sqrt{k/\mu}

where, for a tetrahedral

\ce{AB4}

molecule,

\mu = \frac{3m(A)m(B)}{3m(A)+4m(B)}

. The C–H vibration of methane is 3030.00 cm

^{-1}

and that of the Z-analogue of methane is 2938.45 cm

^{-1}

; the bond enthalpies are 438.4 and 450.2 kJ mol

^{-1}

. Determine the force constant

k

of a C–H bond, estimate

k

of the Z–H bond assuming proportionality to bond enthalpy, and find the atomic mass and symbol of Z.

Step 2 of 4: Hydrogen balance on argyrodite

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\ce{Ag_aY_bS_{a/2+2b}} + b\ce{H2} -> \frac{a}{2}\ce{Ag2S} + b\ce{YS} + b\ce{H2S};\quad n(\ce{H2}) = \frac{pV}{RT} = 8.871\times10^{-3}\ \text{mol}

Analysis

Writing the mineral $\ce{Ag_aY_bS_{a/2+2b}}$ (charge neutrality: $a + 4b = 2(a/2 + 2b)$ ), reduction of the Y(IV) sulfide to YS consumes $b$ mol $\ce{H2}$ per formula unit. From the gas law $n(\ce{H2}) = 100\times10^{3}\times0.295\times10^{-3}/(8.314\times400) = 8.871\times10^{-3}$ mol for 10.0 g of mineral.