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Chapter 19: Problem 24

The element sodium (Na) melts at \(97.8^{\circ} \mathrm{C},\) and its molar enthalpy of fusion is $\Delta H_{\text {fus }}=2.60 \mathrm{~kJ} / \mathrm{mol}$. (a) When molten sodium solidifies to \(\mathrm{Na}(\mathrm{s})\), is \(\Delta S\) positive or negative? (b) Calculate the value of \(\Delta S\) when \(50.0 \mathrm{~g}\) of \(\mathrm{Na}(l)\) solidifies at \(97.8^{\circ} \mathrm{C}\).

Short Answer

Expert verified
(a) When molten sodium solidifies to Na(s), the ΔS is negative. (b) The value of ΔS when 50.0 g of Na(l) solidifies at 97.8°C is -0.0152 kJ/K.

Step by step solution

01

Determine the sign of ΔS

At the melting point, the process of solidification is in equilibrium, which means ΔG = 0. According to the equation ΔG = ΔH - TΔS, at equilibrium, ΔH = TΔS. Since fusion is the process of melting, the enthalpy change for the reverse process, solidification, will have the opposite sign. In this case, ΔH_fus = 2.60 kJ/mol, so ΔH_solidification = -2.60 kJ/mol. We know that ΔH_solidification is negative and T (temperature in Kelvin) is positive. Therefore, to satisfy the equilibrium condition, ΔS must be negative. The answer to part (a) is that ΔS is negative during the solidification of sodium.
02

Convert temperature to Kelvin

In order to calculate ΔS, we need to convert the given temperature in Celsius to Kelvin. To do this, simply add 273.15 to the Celsius value: T = 97.8°C + 273.15 = 370.95 K
03

Calculate ΔS

Now we can use the equilibrium condition ΔH = TΔS to calculate the change in entropy during the solidification process. Rearrange the formula to find ΔS: ΔS = ΔH / T ΔS = (-2.60 kJ/mol) / (370.95 K) ΔS = -0.0070 kJ/(mol·K)
04

Calculate the moles of Na

To find the change in entropy for 50.0 g of Na, we need to calculate the number of moles in that mass. The molar mass of sodium is 22.99 g/mol. Divide the mass by the molar mass: moles_Na = (50.0 g) / (22.99 g/mol) = 2.174 mol
05

Calculate total ΔS

Now we can calculate the total change in entropy for the given mass of Na by multiplying the moles by the change in entropy per mole: ΔS_total = moles_Na × ΔS ΔS_total = (2.174 mol) × (-0.0070 kJ/(mol·K)) ΔS_total = -0.0152 kJ/K The answer to part (b) is that the change in entropy when 50.0 g of Na solidifies at 97.8°C is -0.0152 kJ/K.

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Most popular questions from this chapter

Consider the following process: a system changes from state 1 (initial state) to state 2 (final state) in such a way that its temperature changes from $300 \mathrm{~K}\( to \)400 \mathrm{~K}$. (a) Is this process isothermal? (b) Does the temperature change depend on the particular pathway taken to carry out this change of state? (c) Does the change in the internal energy, \(\Delta E\), depend on whether the process is reversible or irreversible?

Carbon disulfide \(\left(C S_{2}\right)\) is a toxic, highly flammable substance. The following thermodynamic data are available for \(\mathrm{CS}_{2}(I)\) and \(\mathrm{CS}_{2}(g)\) at \(298 \mathrm{~K}\) \begin{tabular}{lcc} \hline & \(\Delta H_{i}(\mathrm{k} / \mathrm{mol})\) & $\Delta G_{i}^{\prime}(\mathrm{kJ} / \mathrm{mol})$ \\ \hline\(C S_{2}(l)\) & 89.7 & 65.3 \\ \(C S_{2}(g)\) & 117.4 & 67.2 \\ \hline \end{tabular} (a) Draw the Lewis structure of the molecule. What do you predict for the bond order of the \(\mathrm{C}-\mathrm{S}\) bonds? \((\mathbf{b})\) Use the VSEPR method to predict the structure of the \(\mathrm{CS}_{2}\) molecule. (c) Liquid \(\mathrm{CS}_{2}\) burns in \(\mathrm{O}_{2}\) with a blue flame, forming \(\mathrm{CO}_{2}(g)\) and \(\mathrm{SO}_{2}(g)\). Write a balanced equation for this reaction. (d) Using the data in the preceding table and in Appendix \(C,\) calculate \(\Delta H^{\circ}\) and \(\Delta G^{\circ}\) for the reaction in part \((c) .\) Is the reaction exothermic? Is it spontaneous at \(298 \mathrm{~K} ?\) (e) Use the data in the table to calculate \(\Delta S^{\circ}\) at $298 \mathrm{~K}\( for the vaporization of \)\mathrm{CS}_{2}(I) .$ Is the sign of \(\Delta S^{\circ}\) as you would expect for a vaporization? (f) Using data in the table and your answer to part (e), estimate the boiling point of \(\mathrm{CS}_{2}(l)\). Do you predict that the substance will be a liquid or a gas at \(298 \mathrm{~K}\) and \(101.3 \mathrm{kPa}\) ?

For a particular reaction, \(\Delta H=30.0 \mathrm{~kJ}\) and $\Delta S=90.0 \mathrm{~J} / \mathrm{K}\(. Assume that \)\Delta H\( and \)\Delta S$ do not vary with temperature. (a) At what temperature will the reaction have \(\Delta G=0 ?\) (b) If \(\mathrm{T}\) is increased from that in part (a), will the reaction be spontaneous or nonspontaneous?

Use data from Appendix \(C\) to calculate the equilibrium constant, \(K,\) and \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\) for each of the following reactions: (a) \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)\) (b) $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g)$ (c) $3 \mathrm{C}_{2} \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{6}(g)$

Most liquids follow Trouton's rule (see Exercise 19.93 ), which states that the molar entropy of vaporization is approximately $88 \pm 5 \mathrm{~J} / \mathrm{mol}-\mathrm{K}$. The normal boiling points and enthalpies of vaporization of several organic liquids are as follows: \begin{tabular}{lcc} \hline & Normal Boiling & \\ Substance & Point \(\left({ }^{\circ} \mathrm{C}\right)\) & $\Delta H_{\text {vap }}(\mathrm{k} / / \mathrm{mol})$ \\ \hline Acetone, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO}\) & 56.1 & 29.1 \\\ Dimethyl ether, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O}\) & -24.8 & 21.5 \\\ Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) & 78.4 & 38.6 \\ Octane, \(\mathrm{C}_{\mathrm{s}} \mathrm{H}_{18}\) & 125.6 & 34.4 \\ Pyridine, \(\mathrm{C}_{5} \mathrm{H}_{\mathrm{S}} \mathrm{N}\) & 115.3 & 35.1 \\\ \hline \end{tabular} (a) Calculate \(\Delta S_{\text {vap }}\) for each of the liquids. Do all the liquids obey Trouton's rule? (b) With reference to intermolecular forces (Section 11.2), can you explain any exceptions to the rule? (c) Would you expect water to obey Trouton's rule? By using data in Appendix \(\mathrm{B}\), check the accuracy of your conclusion. (d) Chlorobenzene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{Cl}\right)\) boils at \(131.8^{\circ} \mathrm{C}\). Use Trouton's rule to estimate $\Delta H_{\text {vap }}$ for this substance.
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