Problem 104 - Chemistry

Question

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 \(\mathrm{kJ} / \mathrm{mol}\) .

Step-by-Step Solution

Answer

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 1: 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

Step 2: 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.

Step 3: 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

Step 4: 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.

Step 5: 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

Key Concepts

Enthalpy Change

Enthalpy change, denoted as \( \Delta H \), is a crucial concept in thermochemistry. It represents the heat exchange in a chemical reaction at constant pressure. When a reaction occurs, it can either absorb or release heat. This change is represented as a positive or negative \( \Delta H \) value, respectively.

In our example, the reaction of sodium with water releases heat, shown by its negative enthalpy change \(-368 \text{kJ}\). This means that energy is released from the system into the surroundings.

Understanding whether \( \Delta H \) is positive or negative helps predict if the reaction will need energy input, like heating, or if it can generate heat on its own.

Heat Capacity

Heat capacity is a measure of the amount of heat needed to change a substance's temperature by one degree Celsius. It is commonly expressed in units of \( \text{J/g} \cdot ^\circ \text{C} \). This property helps us understand how different materials react to adding or removing heat.

For example, in this exercise, the specific heat capacity of the final mixture is given as 4.18 \( \text{J/g} \cdot ^\circ \text{C} \). This tells us how much energy is needed to change the temperature of 1 gram of the mixture by 1°C.

In practice, using heat capacity values allows us to calculate temperature changes, as seen in the formula \( Q = \text{mass} \times C_p \times \Delta T \). This equation relates heat energy (\( Q \)), mass, specific heat capacity (\( C_p \)), and temperature change (\( \Delta T \)).

Enthalpy of Fusion

The enthalpy of fusion, symbolized as \( \Delta H_\text{fusion} \), is the energy required to convert a solid into a liquid at its melting point. It is crucial for processes that involve phase changes, like melting ice.

The value given in our exercise is 6.02 \text{kJ/mol} for ice. This means that to melt one mole of ice, 6.02 kJ of energy is required.

This energy must be provided to overcome the attractive forces between molecules in a solid. In the context of our problem, it's important to know that not enough heat was released by the sodium reaction to melt all the ice. This is because only 2.0 kJ of energy was released, far below the 16.7 kJ needed to fully melt the ice.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products, releasing or absorbing energy in the process. In the example provided, the reaction between sodium and water creates sodium hydroxide and hydrogen gas, also releasing heat.

Reactions can be classified based on how they interact with energy:

• Exothermic: Reactions that release energy, like our sodium reaction.
• Endothermic: Reactions that absorb energy from the surroundings.

Understanding these categories aids in predicting the thermal behavior of reactions.

The sodium reaction, being exothermic, releases energy which impacts whether the ice will melt fully or not. Unfortunately, the heat released was insufficient to melt all the ice, which is crucial for understanding energy flow in chemical transformations.