Question

At the melting point of a solid (or the freezing point of a liquid), the free energies of the solid state and the liquid state are equal, \(\Delta \mathrm{G}=0 .\) Likewise, at the boiling point of a liquid, where there is an equilibrium between the liquid and vapor phases, the free energy is equal in the two states. Calculate the change in entropy for the following process at \(0^{\circ} \mathrm{C}\) if the heat of fusion of \(\mathrm{H}_{2} \mathrm{O}=80 \mathrm{cal} / \mathrm{g} . \mathrm{H}_{2} \mathrm{O}(\mathrm{s}) \rightarrow \mathrm{H}_{2} \mathrm{O}(\ell)\).

Short answer

The change in entropy for the given process at 0°C, when converting 1 gram of ice at its melting point to the liquid state, is approximately 1.22 J/(K·g).

Step-by-step solution

Write the given information and the formula for entropy change.

We are given the heat of fusion of H2O as 80 cal/g and the temperature as 0°C, which we need to convert to Kelvin. Also, we have to calculate the change in entropy, which can be represented as ΔS. The formula to calculate entropy change is: ΔS = \( \frac{q_p}{T} \) Where ΔS is the change in entropy qp is the heat absorbed/released at constant pressure, and T is the absolute (Kelvin) temperature.

Convert the temperature to Kelvin.

We know that 0°C is equivalent to 273.15 K. Therefore, T = 273.15 K.

Calculate the heat absorbed at constant pressure.

The heat of fusion of H2O is given as 80 cal/g. It represents the amount of heat absorbed by 1 gram of ice at its melting point to change into the liquid state without any change in temperature. Therefore, qp = 80 cal/g.

Calculate the entropy change.

Using the formula for entropy change, we can now calculate ΔS: ΔS = \( \frac{q_p}{T} \) = \( \frac{80 \, \text{cal/g}}{273.15 \, \text{K}} \) However, to have a consistent unit system, we need to convert calories into joules, as the standard unit for entropy is J/(K·g). We know that 1 cal = 4.184 J. ΔS = \( \frac{80 \times 4.184 \, \text{J/g}}{273.15 \, \text{K}} \) ≈ 1.22 J/(K·g) Therefore, the change in entropy for the given process at 0°C is approximately 1.22 J/(K·g).

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The fundamental principles of thermodynamics are encapsulated in four laws, with the second law, in particular, defining the concept of entropy. This law states that the entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium. In the context of our exercise, the second law explains why heat flows from the solid ice to its surroundings; it's a natural trend toward increasing disorder or entropy.

Phase Transition

A phase transition is a transformation of a substance from one state of matter to another, such as from solid to liquid, liquid to gas, or vice versa. These transitions occur at characteristic temperatures and pressures, often absorbing or releasing latent heat in the process. The transition at the melting point, where a solid turns into a liquid, involves heat of fusion and is very pertinent to our exercise concerning the melting of ice. During phase transitions, the temperature remains constant while the substance absorbs or releases heat, indicating that the heat energy is used to change the state rather than the temperature.

Heat of Fusion

The heat of fusion is the amount of energy required to change a substance from the solid phase to the liquid phase at its melting point without changing its temperature. This is an essential concept when calculating the entropy change during a phase transition, such as the melting of ice. For instance, each gram of ice requires a specific amount of heat, 80 calories in the case of H2O, to melt into water at its melting point. This heat of fusion must be correctly accounted for in thermodynamic calculations like the one we're discussing.

Entropy Formula

Entropy, symbolized by 'S', is a measure of the disorganization or randomness in a system. The change in entropy (ΔS) can be calculated using the formula ΔS = q_p / T, where q_p represents the heat absorbed or released at constant pressure during the process, and T is the temperature measured on the absolute Kelvin scale. This equation hinges on the understanding that entropy change is directly proportional to the heat absorbed or released but inversely proportional to the temperature. Remember to use consistent units when applying this formula; typically joules for heat and Kelvin for temperature.

Kelvin Temperature Scale

The Kelvin temperature scale is an absolute thermodynamic temperature scale, where absolute zero (0 K) is the point at which all thermal motion ceases in the classical description of thermodynamics. This scale is crucial when doing entropy calculations because the entropy formula requires an absolute temperature measurement. Since degrees Celsius and Kelvin are on different scales, converting Celsius to Kelvin is critical for accurate calculations in thermodynamics. This conversion is simple: T(K) = T(°C) + 273.15. This exact conversion was performed in the exercise to find the temperature in Kelvin for the entropy calculation.