Problem 54 - Chemistry The Science in Context

Question

Which of the following would reach the higher temperature after \(10.00 \mathrm{g}\) of iron \(\left[c_{\mathrm{P}}=25.1 \mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]\) at \(150^{\circ} \mathrm{C}\) is added: 100 mL of water \(\left[d=1.00 \mathrm{g} / \mathrm{mL}, c_{\mathrm{P}}=75.3 \mathrm{J} /\right.\) \(\left.\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]\) or \(200 \mathrm{mL}\) of ethanol \(\left[\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right.\) \(\left.d=0.789 \mathrm{g} / \mathrm{mL}, c_{\mathrm{P}}=113.1 \mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right] ?\)

Step-by-Step Solution

Answer

Answer: To find out which liquid will reach a higher temperature, we need to calculate the temperature change for each liquid, considering that the heat lost by iron is equal to the heat gained by the liquid. After calculating the temperature change for each liquid, we can compare them along with their initial temperatures to find out which one will have a higher final temperature.

Step 1: 1. Calculate the amount of heat transferred from iron

Since heat lost by iron is equal to heat gained by each liquid, we need to find the heat transferred from iron first. We can use the formula: \( q = m \times c_{p} \times \Delta T \) Where q is the heat transferred, m is the mass, \(c_{p}\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. For iron: \(q_{iron} = m_{iron} \times c_{p_{iron}} \times \Delta T_{iron} \)

Step 2: 2. Calculate the moles of each liquid

To compare water and ethanol, we'll require their properties. First, we'll find the number of moles using the densities given. \(i\). For water, mass = volume × density and moles = mass / molar mass \( n_{water} = \frac{100mL \times 1.00g/mL}{18.02g/mol} \) \(ii\). For ethanol, mass = volume × density \( n_{ethanol} = \frac{200mL \times 0.789g/mL}{46.07g/mol} \)

Step 3: 3. Calculate heat gained by each liquid and the final temperature.

Now, we will calculate the heat gained by each liquid using the heat transferred from iron, moles, and specific heat capacities. Heat gained by water: \(q_{water} = n_{water} \times c_{p_{water}} \times \Delta T_{water} \) Heat gained by ethanol: \(q_{ethanol} = n_{ethanol} \times c_{p_{ethanol}} \times \Delta T_{ethanol} \) Since the heat lost by iron is equal to the heat gained by the liquid, we can equate the two expressions and solve for the temperature change for each liquid. For water: \(\Delta T_{water} = \frac{q_{iron}}{n_{water} \times c_{p_{water}}} \) For ethanol: \(\Delta T_{ethanol} = \frac{q_{iron}}{n_{ethanol} \times c_{p_{ethanol}}} \)

Step 4: 4. Compare to find which liquid reaches a higher temperature.

Now, we will compare the calculated temperature changes, adding these values to their initial temperatures, to find which liquid will have a higher final temperature. (Assuming they start at the same temperature.) If \(\Delta T_{water} + T_{initial} > \Delta T_{ethanol} + T_{initial}\), then water reaches a higher temperature. If \(\Delta T_{ethanol} + T_{initial} > \Delta T_{water} + T_{initial}\), then ethanol reaches a higher temperature.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a vital concept in thermodynamics that refers to the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin).
It's denoted by the symbol \(c_p\). It acts as a measure of how much heat energy a substance can store. For instance, substances with a higher specific heat capacity, like water, can absorb more heat before their temperature rises significantly.
In the context of the exercise, the specific heat capacity is used to calculate the heat gained by water and ethanol when iron is added.

• The specific heat capacity of water is \(75.3\, \text{J/mol} \cdot ^\circ \text{C}\), meaning it takes 75.3 joules to raise 1 mole of water by 1 degree Celsius.
• Ethanol's specific heat capacity is \(113.1\, \text{J/mol} \cdot ^\circ\text{C}\), indicating that it requires even more heat to achieve the same temperature change.

Understanding this helps explain why different substances heat up at different rates when the same amount of heat is added.

Heat Transfer

Heat transfer describes the movement of thermal energy from one object to another due to temperature differences. In the exercise, when iron at a high temperature is added to water or ethanol, heat is transferred from iron to the liquid, causing the liquid's temperature to rise.
The fundamental principle governing this process is the conservation of energy. This principle asserts that the heat lost by iron must be equal to the total heat gained by the water and ethanol.
Mathematically, this is expressed as \( q_{\text{iron}} = q_{\text{water}} = q_{\text{ethanol}} \). You can calculate \( q \), the amount of heat transfer, using the formula:

• \( q = m \times c_{p} \times \Delta T \), where \( m \) is the mass, \( c_{p} \) is the specific heat capacity, and \( \Delta T \) is the temperature change.

This equation helps us understand how much energy is transferred and how it affects the temperature of materials involved.

Calculating Moles

Calculating moles is essential for determining how substances react and for thermal calculations. A mole is a standard unit in chemistry that measures the amount of substance. It gives us a way to relate mass to particle numbers.
In the problem, we need to find out how many moles of water and ethanol there are to determine how heat transfer affects them.
The formula to calculate moles \( n \) is:

• \( n = \frac{\text{mass}}{\text{molar mass}} \)

Water and ethanol masses are calculated using their volume and density, and the result is then used to find the moles.
For water: the mass is found by \(100 \text{mL} \times 1.00\,\text{g/mL}\), and then moles by dividing by water's molar mass \(18.02\,\text{g/mol}\).
For ethanol: the mass is \(200 \text{mL} \times 0.789\,\text{g/mL}\), with moles calculated by dividing by the molar mass \(46.07\,\text{g/mol}\).
This crucial step helps in understanding how temperature changes will affect different substances when heat is exchanged.