Question

Calculate the entropy change that occurs when 0.50 mol of ice is converted to liquid water at \(0^{\circ} \mathrm{C}\) in a reversible process. \(\left(q_{\text {fus }}=333 \mathrm{J} / \mathrm{g}\right).\)

Short answer

The entropy change is approximately 10.97 J/K.

Step-by-step solution

Convert Mass to Moles

First, we need to convert the mass of ice to moles. The molar mass of water is approximately 18.0 g/mol. Since we have 0.50 mol of ice:\[ m = n \times M \]\[ m = 0.50 \text{ mol} \times 18.0 \text{ g/mol} = 9.0 \text{ g} \]So, we have 9.0 grams of ice.

Calculate Heat of Fusion

Calculate the total heat required for the phase change using the heat of fusion formula:\[ q = m \times q_{\text{fus}} \]\[ q = 9.0 \text{ g} \times 333 \text{ J/g} = 2997 \text{ J} \]Thus, 2997 J is required to convert the ice to liquid water.

Convert Temperature to Kelvin

As entropy changes are most accurately calculated in Kelvin:\[ T = 0^{\circ}C + 273.15 = 273.15 \text{ K} \]

Use Entropy Change Formula

The change in entropy for a reversible phase change is given by:\[ \Delta S = \frac{q}{T} \]Thus,\[ \Delta S = \frac{2997 \text{ J}}{273.15 \text{ K}} \approx 10.97 \text{ J/K} \]

Final Entropy Change

The entropy change when 0.50 mol of ice is converted to liquid water at \(0^{\circ}C\) in a reversible process is approximately 10.97 J/K.

Key concepts

Heat of Fusion

The heat of fusion is a critical concept to understand when discussing phase transitions, specifically when converting a solid to a liquid. It represents the amount of energy required per unit mass to change a substance from a solid to a liquid at its melting point, without changing its temperature. In this exercise, the heat of fusion is given as 333 J/g, which is the specific energy needed to melt ice into water at 0°C.

To find the total energy (or heat) needed for the phase transition, we multiply the mass of the substance by the heat of fusion. This provides the energy required for the entire sample of the ice to turn into water. This calculation is essential as it sets the groundwork for understanding changes in entropy, given that these changes occur during this heat-dependent process.

Molar Mass Conversion

Understanding molar mass conversion is vital in chemistry. The molar mass of a substance is the mass of one mole of its molecules. For water, the molar mass is approximately 18.0 g/mol.

In this problem, we are given 0.50 moles of ice, which we need to convert to grams to use in further computations, like calculating the total heat of fusion. Using the formula:

• Mass (m) = Number of moles (n) \( \times \) Molar mass (M)

We substitute the known values to find the mass in grams. This conversion ensures we have the correct unit of measure, enabling more accurate calculations in the subsequent steps, such as finding the total energy required for the phase change.

Kelvin Temperature

The Kelvin scale is an absolute temperature scale vital in thermodynamics and scientific computations. Unlike Celsius or Fahrenheit, Kelvin starts at absolute zero, which is the theoretically lowest possible temperature.

Converting degrees Celsius to Kelvin is straightforward: you simply add 273.15. In this exercise, the temperature is initially 0°C, which converts to 273.15 K.

Using Kelvin for our equations is important because entropy and many thermodynamic properties are directly linked to an absolute temperature scale. This gives a more consistent and universal standard of measurement when dealing with the laws of thermodynamics, such as calculating the change in entropy in reversible processes.

Reversible Process

A reversible process is an idealized process that always remains in equilibrium. This means that the system changes in such a way that the process can be reversed by an infinitesimal change in some property.

In practice, this isn't possible, but it's a useful model for studying thermodynamic properties, like entropy. The significance of the reversible process in our exercise is that it allows us to simplify computations when calculating entropy, using the equation:

• \( \Delta S = \frac{q}{T} \)

'\( q \)' refers to the heat added in the process, while '\( T \)' is the absolute temperature in Kelvin. Hence, employing the concept of a reversible process ensures a more accessible and systematic approach to determining the change in entropy when ice turns to liquid water at 0°C.