Question

A \(0.250-\mathrm{g}\) 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 \(2 \mathrm{Na}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{NaOH}(a q)+\mathrm{H}_{2}(g) \quad \Delta H=-368 \mathrm{~kJ}\) 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

The ice will not melt completely as the heat released by the reaction of sodium metal with water is not enough to melt all the ice. The final temperature of the mixture will be 4.43°C.

Step-by-step solution

Calculate the moles of sodium metal

Given that the mass of sodium metal is 0.250 g, we can find the moles of sodium metal by dividing the mass by its molar mass (23 g/mol): \[moles\:of\:Na = \frac{0.250\:g}{23\:g/mol} = 0.0109\:mol\]

Calculate the heat released by the reaction

Now, we can calculate the heat released by the reaction using the moles of sodium metal and the given reaction enthalpy: \[q = (moles\:of\:Na) \times (\frac{1\:mol\:H_2}{2\:mol\:Na}) \times \Delta H\] \[q = (0.0109\:mol) \times \frac{1}{2} \times (-368\:kJ) = -1.859\: kJ\]

Calculate the heat required to melt the ice

The heat required to melt the ice can be calculated using the enthalpy of fusion and the moles of water present in the ice: \[ Heat\:required = (moles\:of\:H_2O)\times \Delta H_{fusion}\] Since 1 mol of water has a mass of 18 g, we have: \[moles\:of\:H_2O = \frac{50.0\:g}{18\:g/mol} = 2.778\:mol\] \[Heat\:required = (2.778\:mol) \times (6.02 \: kJ/mol) = 16.716\:kJ\]

Determine if the ice will melt

Since the heat released by the reaction (-1.859 kJ) is less than the heat required to melt the ice (16.716 kJ), the ice does not melt completely.

Calculate the final temperature of the mixture

Now we have to calculate the final temperature of the mixture. We know that the specific heat capacity of the final mixture is 4.18 J/g°C. Since the heat released from the reaction will be absorbed by the ice and water mixture, we can set up the following equation: \[q_{absorbed} = (mass_{ice} + mass_{water})\times c_{mixture}\times \Delta T\] \[\Delta T = \frac{-q_{reaction}}{(mass_{ice} + mass_{water})\times c_{mixture}} \] \[ \Delta T = \frac{1.859 \times 10^3\:J}{(50.0\:g + 50.0\:g) \times 4.18 \: J/g\cdot{ }^{\circ}C} = 4.43^{\circ}\:C\] Since the initial temperature is 0°C, the final temperature is: \[ T_{final} = T_{initial} + \Delta T = 0^{\circ}C + 4.43^{\circ}C = 4.43{ }^{\circ}\mathrm{C}\] Thus, the answer is that the ice will not melt, and the final temperature of the mixture will be 4.43°C.

Key concepts

Enthalpy of Reaction

Enthalpy of reaction is the heat change during a chemical reaction. In our exercise, the reaction between sodium metal and water releases energy. This is indicated by a negative enthalpy change \( \Delta H = -368 \, \mathrm{kJ} \). The negative sign means the reaction is exothermic, releasing heat into the surroundings.

To calculate the total heat released, we first find the moles of sodium used in the reaction. With \( 0.250 \, \mathrm{g} \) of sodium and its molar mass \( 23 \, \mathrm{g/mol} \), we determine the moles as \( 0.0109 \, \mathrm{mol} \). Multiplying these moles by the enthalpy change per mole for the balanced reaction yields the heat released: \( -1.859 \, \mathrm{kJ} \). This value tells us how much energy is released when the reaction occurs.

Enthalpy of Fusion

Enthalpy of fusion is the energy required to change a solid into a liquid at its melting point. For ice, this is \( 6.02 \, \mathrm{kJ/mol} \). It represents the energy needed to melt 1 mole of ice without changing the temperature.

The problem involves calculating whether enough heat is available to melt the ice. With ice weighing \( 50.0 \, \mathrm{g} \), we find the moles: \( \frac{50.0 \, \mathrm{g}}{18 \, \mathrm{g/mol}} = 2.778 \, \mathrm{mol} \). The energy required to melt this amount of ice is \( 16.716 \, \mathrm{kJ} \).

Since the reaction releases less energy than needed to melt the ice, not all the ice will turn into water.

Specific Heat Capacity

Specific heat capacity is the amount of heat energy required to raise the temperature of 1 gram of a substance by 1°C. The specific heat capacity of water in our scenario is \( 4.18 \, \mathrm{J/g°C} \), a standard value for water, which also applies to the ice-water mixture.

In our calculation, the combined mass of the ice and water is \( 100.0 \, \mathrm{g} \). Using the heat absorbed equation \( q = mc\Delta T \), where \( m \) is mass and \( c \) is specific heat, we solve for \( \Delta T \) to find the temperature change caused by the heat released from the reaction. This change is \( 4.43°C \), which adds to the initial temperature to yield a final temperature.

Heat Transfer in Reactions

Heat transfer in reactions involves the movement of thermal energy from the reaction to the surrounding environment. Here, the exothermic reaction between sodium and water releases heat that is absorbed by the ice-water mixture, potentially causing a temperature rise.

In this exercise, the total energy released by the reaction was insufficient to melt all the ice. Instead, the heat transfer raises the temperature of the mixture. We calculate the change in temperature \( \Delta T \) by dividing the heat released by the total mass and specific heat capacity of the mixture.

Ultimately, this illustrates how energy conservation and transfer play crucial roles in determining the outcomes of physical and chemical processes, such as changes in temperature or phase.