Question

You have a 1.00 -L sample of hot water \(\left(90.0^{\circ} \mathrm{C}\right)\) sitting open in a \(25.0^{\circ} \mathrm{C}\) room. Eventually the water cools to \(25.0^{\circ} \mathrm{C}\) while the temperature of the room remains unchanged. Calculate \(\Delta S_{\mathrm{sur}}\) for this process. Assume the density of water is 1.00 \(\mathrm{g} / \mathrm{cm}^{3}\) over this temperature range, and the heat capacity of water is constant over this temperature range and equal to 75.4 \(\mathrm{J} / \mathrm{K} \cdot\) mol.

Short answer

The change in entropy of the surroundings, \(\Delta S_{\text{sur}}\), for the process of hot water cooling to room temperature is 871.6 J/K.

Step-by-step solution

Find the number of moles of water

First, you need to find the number of moles of water in the 1.00 L sample. Use the density of water, 1.00 g/cm³, to do this. Since 1 L = 1000 cm³, you have: Mass of water = Volume × Density = 1000 cm³ × 1.00 g/cm³ = 1000 g Now, find the number of moles of water, knowing that the molar mass of water is 18.02 g/mol: Number of moles = Mass / Molar mass = 1000 g / 18.02 g/mol = 55.49 mol

Calculate the heat exchanged

Now, calculate the amount of heat released by the water as it cools down from 90.0°C to 25.0°C. Use the heat capacity of water and the temperature difference: \(q = n \times C \times \Delta T\) \(q = 55.49 \mathrm{mol} \times 75.4 \mathrm{\frac{J}{K \cdot mol}} \times (25.0 ^\circ \mathrm{C} - 90.0 ^\circ \mathrm{C})\) \(q = 55.49 \mathrm{mol} \times 75.4 \mathrm{\frac{J}{K \cdot mol}} \times -65.0 \mathrm{K}\) \(q = -259910.3 \mathrm{J}\) The heat exchanged (q) is negative because it was released to the surroundings by the water sample.

Calculate the change in entropy of the surroundings

Finally, calculate the change in entropy of the surroundings with the formula: \(\Delta S_{\text{sur}} = -\frac{q}{T}\) Since the temperature of the room remains constant at 25.0°C, you can use that in the formula, but you need to convert it to Kelvin first: 25.0°C + 273.15 = 298.15 K \(\Delta S_{\text{sur}} = -\frac{-259910.3 \mathrm{J}}{298.15 \mathrm{K}}\) \(\Delta S_{\text{sur}} = 871.6 \mathrm{\frac{J}{K}}\) The change in entropy of the surroundings, \(\Delta S_{\text{sur}}\), is 871.6 J/K.

Key concepts

Water Cooling

Water cooling is a natural process where the water's temperature decreases until it reaches equilibrium with its surroundings. In this scenario, the sample starts at 90°C and gradually adjusts to the room temperature of 25°C. This process is crucial for understanding thermodynamics, as it exemplifies energy transfer. The cooling of water involves releasing heat to the surroundings until an equal temperature is reached, demonstrating the principle of heat exchange.
The cooling effect can be observed not only in everyday experiences but also in industrial settings where controlled cooling is necessary for various processes. It is also essential for understanding concepts such as entropy and heat capacity, which dictate energy distribution in any given system. Understanding water cooling helps students grasp larger principles of physics and chemistry, grounding theoretical knowledge in practical, observable phenomena.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change a substance's temperature by a certain amount. In this example, the heat capacity of water is given as 75.4 J/K·mol. This value tells us how much energy, in joules, is necessary to raise the temperature of one mole of water by one degree Kelvin.
Heat capacity is an intrinsic property, meaning it is specific to each material. For water, known for its relatively high heat capacity, this feature explains why it can absorb or release large amounts of heat without experiencing quick temperature changes.

• The high heat capacity of water is crucial for moderating Earth's climate and biological systems.
• When calculating thermal changes in systems, understanding heat capacity allows accurate predictions of how substances will react to energy input or removal.

By grasping heat capacity, students can better understand and predict the behavior of substances under various thermal conditions.

Number of Moles

The number of moles is a fundamental concept in chemistry that represents a specific quantity of substance. It helps in quantifying the amount of molecules or atoms in a given sample. For the water sample in this exercise, the number of moles is calculated using its mass and the molar mass of water, 18.02 g/mol. First, the mass of water is determined as 1000 g since we know 1 L of water weighs about 1000 g.
Next, dividing this mass by the molar mass of water gives us approximately 55.49 moles. Understanding moles is essential for balancing chemical equations, calculating concentrations, and performing various stoichiometric calculations in chemistry.

• Moles bridge the gap between atomic-scale properties and macroscopic measurements we can observe.
• They simplify complex interactions into understandable proportions.

Recognizing how to compute and use moles empowers students to explore deeper chemical relationships and interactions.

Heat Exchange

Heat exchange describes the energy transfer between a system and its surroundings. In our example, heat transfer occurs when hot water releases thermal energy into the cooler surrounding air, resulting in the water's temperature dropping. The magnitude of this heat exchange can be calculated using the formula:\( q = n \times C \times \Delta T \)Here, 'q' represents the heat exchanged, 'n' stands for the number of moles, 'C' is the heat capacity, and \( \Delta T \) signifies the temperature change.
In this scenario, the negative sign of \( q \) confirms that energy leaves the water as it cools. Understanding heat exchange is pivotal in thermodynamics, where energy conservation and transformation are key principles.

• Heat exchange impacts various processes, from everyday cooking to industrial cooling systems.
• It illustrates the flow of energy and contributes to understanding entropy—a measure of disorder within a system.

Mastering this concept allows students to comprehend energy transfer processes, equipping them with insights into both simple and sophisticated thermal systems.