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In chemistry, calculating the molar mass is the first crucial step in many calculations such as finding the moles present in a given mass. The molar mass of an element, like sodium (\(\mathrm{Na}\)), is essentially the mass of one mole of the element. On the periodic table, it is expressed in grams per mole. Sodium has a molar mass of approximately \(22.99 \, \text{g/mol}\).
To find out how many moles are in a sample, use the formula:

• Moles of substance = \(\frac{\text{Mass of substance}}{\text{Molar mass of substance}}\)

For example, if you have \(50.0 \, \text{g} \) of sodium, divide this by \(22.99 \, \text{g/mol}\) to find the moles. This gives approximately \(2.174 \, \text{mol}\), which will be crucial for further calculations like heat calculations.

Heat capacity is a property that describes how much heat energy is required to change the temperature of a given amount of substance by 1 degree Kelvin (or Celsius). In this example with solid sodium, the heat capacity is given as \(28.2 \, \mathrm{J} / (\mathrm{K} \cdot \text{mol})\).
This means it requires 28.2 joules of energy to increase the temperature of 1 mole of sodium by 1 Kelvin. It is a very useful property in calculating the heat needed for a temperature change.
Consider the temperature change from \(25.0^{\circ} \text{C} \) to \(97.8^{\circ} \text{C}\). The temperature change (\(\Delta T\)) is therefore \(72.8 \, \text{K}\). By multiplying the number of moles, heat capacity, and the temperature difference, you can calculate the required heat. In our case:
\[\text{Heat} (q_1) = n \times C \times \Delta T\]
\[q_1 = 2.174 \, \text{mol} \times 28.2 \, \frac{\text{J}}{\text{K} \cdot \text{mol}} \times 72.8 \, \text{K} \approx 4460.2 \, \text{J}\]

Phase change involves the conversion of a substance from one state of matter to another, such as solid to liquid. This requires energy known as the heat of fusion for melting. This special kind of heat exchange doesn't involve a temperature change but a state change.
For sodium, the heat of fusion is \(2.60 \, \mathrm{kJ/mol}\), or \(2600 \, \mathrm{J/mol}\).
To calculate the heat needed for this transition, the formula is:

• Heat required = \(n \times \Delta H_f\)

Where \(n\) is the number of moles and \(\Delta H_f\) is the heat of fusion.
For \(2.174 \, \text{mol}\) of sodium, plug in the values to find:
\[q_2 = 2.174 \, \text{mol} \times 2600 \, \frac{\text{J}}{\text{mol}} = 5652.4 \, \text{J}\]

The specific heat formula is essential for calculating the heat required to change the temperature of a substance. This formula connects the amount of heat added or removed to the mass, the specific heat capacity, and the temperature change. The formula is:

• \(q = m \cdot C_s \cdot \Delta T\)

Where:

• \(q\) is the heat added or removed
• \(m\) is the mass of the substance
• \(C_s\) is the specific heat capacity (per gram, different from molar heat capacity)
• \(\Delta T\) is the change in temperature

In problems where you have the molar heat capacity, the moles are multiplied instead of mass, using the heat capacity given in \(\text{J/mol}\cdot \text{K}\). For sodium, converting the energy calculated from the heat capacity into total heat by considering both the temperature change and phase change adds up to the overall heat requirement. The total \(q\) was calculated to be approximately \(10113 \, \text{J}\). This includes both raising the temperature of solid sodium and melting it into a liquid.