Problem 148 - General Chemistry

Question

How much heat must be added to \(25.0 \mathrm{~g}\) of solid sodium, Na, at \(25.0^{\circ} \mathrm{C}\) to give the liquid at its melting point, \(97.8^{\circ} \mathrm{C}\) ? The heat capacity of solid sodium is \(28.2 \mathrm{~J} /(\mathrm{K} \cdot \mathrm{mol})\), and its heat of fusion is \(2.60 \mathrm{~kJ} / \mathrm{mol}\).

Step-by-Step Solution

Answer

5058.8 J

Step 1: Calculate the number of moles of sodium

First, determine the number of moles of sodium (\( ext{Na}\)) using its molar mass. The molar mass of sodium is approximately \(22.99 \, \text{g/mol}\). Use the formula:\[\text{moles} = \frac{\text{given mass (g)}}{\text{molar mass (g/mol)}}\]Given mass = \(25.0 \, \text{g}\), thus:\[\text{moles} = \frac{25.0}{22.99} \approx 1.087 \, \text{mol}\]

Step 2: Heat required to raise temperature of solid sodium

Next, calculate the heat needed to raise the temperature of solid sodium from \(25.0^{\circ} \text{C}\) to its melting point \(97.8^{\circ} \text{C}\) using the formula:\[q_{1} = n \cdot C_{p} \cdot \Delta T\]Where \(n\) is the number of moles, \(C_{p}\) is the molar heat capacity, and \(\Delta T\) is the change in temperature.\[\Delta T = 97.8 - 25.0 = 72.8 \, ^\circ \text{C} = 72.8 \, \text{K} \]\[q_{1} = 1.087 \, \text{mol} \times 28.2 \, \text{J/mol \cdot K} \times 72.8 \, \text{K} = 2232.6 \, \text{J}\]

Step 3: Heat required for the phase change

Calculate the heat needed for the phase change from solid to liquid at the melting point using the heat of fusion:\[q_{2} = n \cdot \text{heat of fusion}\]Given heat of fusion is \(2.60 \, \text{kJ/mol}\), convert to Joules: \(2.60 \, \text{kJ/mol} = 2600 \, \text{J/mol}\).\[q_{2} = 1.087 \, \text{mol} \times 2600 \, \text{J/mol} = 2826.2 \, \text{J}\]

Step 4: Total heat required

Add the heat required to raise the temperature (\(q_{1}\)) and the heat required for the phase change (\(q_{2}\)) to find the total heat:\[q_{\text{total}} = q_{1} + q_{2} = 2232.6 \, \text{J} + 2826.2 \, \text{J} = 5058.8 \, \text{J}\]

Key Concepts

Molar Heat Capacity

Molar heat capacity is a crucial aspect of understanding how substances absorb heat. It refers to the amount of heat needed to change the temperature of one mole of a substance by one Kelvin. In the case of sodium, the molar heat capacity is provided as 28.2 J/(K·mol). This value helps in calculating the energy required to escalate the temperature of a mole of sodium by one Kelvin.
\[C_{p} = 28.2 \, \frac{\text{J}}{\text{mol} \cdot \text{K}}\]
When calculating the heat (q) needed to raise the temperature, the equation:

• \[q = n \cdot C_{p} \cdot \Delta T\]

becomes useful. Here, \(n\) is the number of moles, and \(\Delta T\) is the temperature change in Kelvin.
Molar heat capacity gives insight into the thermal properties of a substance and is essential for calculating energy changes during heating or cooling processes.

Heat of Fusion

The heat of fusion is the amount of energy required to change a substance from solid to liquid at its melting point, without changing its temperature. For sodium, this value is provided as 2.60 kJ/mol. This signifies that 2.60 kJ of energy is necessary to melt one mole of sodium entirely.
To convert this energy to Joules, which is often needed in calculations along with molar heat capacity and other parameters, we have:

• \(2.60 \, \text{kJ/mol} = 2600 \, \text{J/mol}\)

During phase change calculations, the formula:

• \[q = n \cdot \text{heat of fusion}\]

becomes applicable, where \(n\) is the number of moles. Heat of fusion is fundamental in determining the energy required during melting processes, which is vital in various physical and chemical applications.

Phase Change

A phase change refers to the transformation from one state of matter to another, such as from solid to liquid or vice versa. In this context, it involves melting—the change of sodium from solid to liquid.
It's important to note that during a phase change, temperature remains constant, unlike during temperature changes where energy increases temperature. The heat of fusion plays a key role in phase changes, indicating the energy needed to overcome molecular attractions within a solid, allowing molecules to move freely as a liquid.
The phase change can be summed up using:

• \[q = n \cdot \text{heat of fusion}\]

This expression shows how the number of moles and the heat of fusion determine the energy absorbed or released during a phase change process. Understanding phase changes is crucial for clear insights into energy considerations in thermodynamics.

Temperature Change

Temperature change involves adjusting the temperature of a substance without altering its phase. It’s represented by the difference in the initial and final temperatures, termed \(\Delta T\). For sodium, the initial temperature is 25.0°C, and the final temperature is its melting point, 97.8°C. The difference \(\Delta T = 72.8 \, ^\circ \text{C} = 72.8 \, \text{K}\) is used for calculations.
Molar heat capacity allows us to calculate the energy required based on temperature changes using the formula:

• \[q = n \cdot C_{p} \cdot \Delta T\]

Here, \(n\) represents the number of moles, \(C_{p}\) is the molar heat capacity, and \(\Delta T\) is the temperature change.
Understanding temperature changes is pivotal as it helps determine how much energy is needed to achieve desired temperature levels in various chemical and physical processes.