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Chapter 10: Problem 104

A 0.250 -g chunk of sodium metal is cautiously dropped into a mixture of 50.0 \(\mathrm{g}\) water and 50.0 \(\mathrm{g}\) ice, both at \(0^{\circ} \mathrm{C}\) . The reaction is $$ 2 \mathrm{Na}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{NaOH}(a q)+\mathrm{H}_{2}(g) \quad \Delta H=-368 \mathrm{kJ} $$ Assuming no heat loss to the surroundings, will the ice melt? Assuming the final mixture has a specific heat capacity of 4.18 $\mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C}$ , calculate the final temperature. The enthalpy of fusion for ice is 6.02 \(\mathrm{kJ} / \mathrm{mol}\) .

Short Answer

Expert verified
The ice will not melt completely, as the heat released by the reaction (2.0 kJ) is less than the heat required to melt the ice (16.7 kJ). The final temperature of the mixture is approximately -4.77 °C.

Step by step solution

01

Determine moles of sodium

The first step is to convert the mass of sodium (0.250 g) into moles. We can do this using the molar mass of sodium, which is approximately 23 g/mol: moles of Na = (0.250 g) / (23 g/mol) ≈ 0.01087 mol
02

Calculate heat released by the reaction

Next, we need to calculate the heat released by the reaction using the enthalpy change (∆H) and moles of sodium (n): Q = ΔH × (moles of Na / 2) Q = -368 kJ × (0.01087 mol / 2) Q = -2.0 kJ The negative sign indicates that heat is being released.
03

Calculate heat required to melt the ice

Now, we need to convert the mass of ice (50.0 g) to moles and calculate the heat required to melt the ice: moles of ice = (50.0 g) / (18.015 g/mol) ≈ 2.776 mol Q_required = ΔH_fusion × moles of ice Q_required = 6.02 kJ/mol × 2.776 mol Q_required ≈ 16.7 kJ
04

Compare heat released to heat required

The heat released by the reaction is 2.0 kJ, while the heat required to melt the ice is 16.7 kJ. Since the heat released is less than the heat required, not all of the ice will melt.
05

Calculate final temperature

To calculate the final temperature, we'll assume the water and ice mixture has a specific heat capacity of 4.18 J/g°C: Q = mass × Cp × ΔT ΔT = Q / (mass × Cp) First, we need to convert the heat to J by multiplying by 1000 (1 kJ = 1000 J): Q = -2.0 kJ × 1000 = -2000 J Next, we'll plug in the values and solve for ΔT: ΔT = (-2000 J) / ((50 g + 50 g) × 4.18 J/g°C) ΔT ≈ -4.77 °C The final temperature of the mixture is: T_final = 0°C + (-4.77 °C) = -4.77 °C

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