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

The ice will not melt as the heat released by the reaction (-2.0022 kJ) is less than the heat required to melt the entire ice (16.67 kJ). The final temperature of the mixture will be approximately 4.79°C.

Step-by-step solution

1. Calculate the number of moles of sodium

First, we need to determine the number of moles of sodium. We can use the mass of sodium and the molar mass of sodium to do this. Sodium has a molar mass of 22.99 g/mol. The equation we will use is: Moles of sodium = mass of sodium / molar mass of sodium Plug in the values: Moles of sodium = 0.250 g / 22.99 g/mol = 0.01087 mol

2. Calculate heat released by the reaction

Now we can use the enthalpy of the reaction and the number of moles of sodium to calculate the heat released by the reaction. The equation to do this is: Heat released by reaction = moles of sodium × (∆H / 2) Here, we divide by 2 because the reaction produces 2 moles of NaOH and one mole of H2 gas for every 2 moles of solid sodium. Plug in the values: Heat released by reaction = 0.01087 mol × (-368 kJ/(2 mol)) = -2.0022 kJ

3. Calculate the heat needed to melt the ice

Now, let's calculate the heat required for all the ice to melt using the enthalpy of fusion for ice and the mass of the ice. The equation to do this is: Heat required to melt ice = mass of ice × (enthalpy of fusion / molar mass of water) The molar mass of water is 18.015 g/mol. Plug in the values: Heat required to melt ice = 50.0 g × (6.02 kJ/mol / 18.015 g/mol) = 16.67 kJ

4. Determine if the ice will melt

Now, let's compare the heat produced by the reaction to the heat required to melt the ice. If the heat produced by the reaction is greater than or equal to the heat required to melt the ice, the ice will melt. In this case, we have: Heat released by reaction = -2.0022 kJ Heat required to melt ice = 16.67 kJ Since the heat released by reaction is less than that required to melt the ice, the ice will not melt.

5. Calculate Final Temperature

Now let's calculate the final temperature of the mixture assuming the final mixture has a specific heat capacity of 4.18 J/g·°C. To do this, we can use the equation: Heat gained by mixture = Heat released by reaction The mass of the entire mixture is 100 g (50 g water + 50 g ice). The specific heat capacity of the mixture is given as 4.18 J/g·°C. Therefore, the equation for the heat gained by the mixture becomes: mass of mixture × specific heat capacity × temperature change = 2.0022 kJ Plug in the values: (100 g) × (4.18 J/g·°C) × temperature change = 2002.2 J Now, we can solve for the temperature change: temperature change = (2002.2 J) / (100 g × 4.18 J/g·°C) = 4.788°C Since the initial temperature of the mixture was 0°C, the final temperature will be: Final temperature = Initial temperature + temperature change = 0°C + 4.788°C ≈ 4.79°C

Key concepts

Enthalpy of Reaction

When understanding thermochemistry, one key concept is the enthalpy of reaction, which is essentially the heat change that results from a chemical reaction. In our original exercise, the reaction given is:

• \[2 \mathrm{Na}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2\mathrm{NaOH}(aq)+\mathrm{H}_{2}(g)\quad \Delta H=-368 \mathrm{~kJ}\]

This means that for every two moles of sodium reacting, 368 kJ of heat is released, making it an exothermic reaction (since \(\Delta H\) is negative).
The quantity of heat exchanged in a reaction can be calculated using the equation:

• Heat released = moles of reactant × (\(\Delta H\)/number of moles in reaction equation).

For our situation, since the calculated moles of sodium are 0.01087, the heat released comes out to be approximately -2.0022 kJ. This showcases how enthalpy values are used to quantify the energy changes in reactions.

Enthalpy of Fusion

The enthalpy of fusion is another fundamental thermochemical concept that defines the amount of energy required to change a substance from the solid phase to the liquid phase without changing its temperature.
In simpler terms, it tells us the heat needed for melting.
For water (ice), the enthalpy of fusion is given as 6.02 kJ/mol.
Given the mass of ice in the problem is 50.0 g, we use formula:

• Heat required to melt ice = mass of ice × (enthalpy of fusion/molar mass of water)
• Calculate using 50.0 g and 18.015 g/mol as the molar mass of water.

This results in a calculated heat requirement of approximately 16.67 kJ to melt all the ice. Notice that this value is significantly higher than our previously calculated reaction heat (-2.0022 kJ), indicating that the ice does not completely melt in the reaction because it needs much more heat for full melting than what is produced.

Specific Heat Capacity

Specific heat capacity is a term that defines how much heat is required to raise the temperature of a given mass of a substance by one degree Celsius. Here, the specific heat capacity of the final mixture after the reaction is noted as 4.18 J/g°C.
This means that to increase the temperature of one gram of the mixture by one degree Celsius, 4.18 Joules of energy is required.
In our scenario:

• The total mass of the mixture (water + ice) is 100 g.
• The heat released by the reaction is 2002.2 J (converted from kJ).
• Using the formula for heat transfer: mass × specific heat capacity × temperature change = heat released.

Substituting values gives the temperature change upon reaction to be about 4.79°C, resulting in a final mixture temperature of approximately 4.79°C, as the starting point was 0°C (frozen state). Specific heat capacity is crucial because it determines how much the temperature of a material can change when energy is added or removed.