Question

Ice at \( -5^\circ C \) is heated to become vapor with temperature of \( 110^\circ C \) at atmospheric pressure. The entropy change associated with this process can be obtained from:

• A. \( \int_{268 \, \text{K}}^{383 \, \text{K}} C_p \, dT + \frac{\Delta H_{\text{melting}}}{273} + \frac{\Delta H_{\text{boiling}}}{373} \)
• B. \( \int_{268 \, \text{K}}^{273 \, \text{K}} \frac{C_{p,m}}{T} \, dT + \frac{\Delta H_m \, \text{fusion}}{T_f} + \frac{\Delta H_m \, \text{vaporisation}}{T_b} \)
• C. \( \int_{268 \, \text{K}}^{373 \, \text{K}} C_p \, dT + q_{\text{rev}} \)
• D. \( \int_{268 \, \text{K}}^{273 \, \text{K}} C_p \, dT + \frac{\Delta H_m \, \text{fusion}}{T_f} + \frac{\Delta H_m \, \text{vaporisation}}{T_b} + \int_{373 \, \text{K}}^{383 \, \text{K}} C_p \, dT \)

Hint:

When calculating entropy change in heating processes, remember to account for both temperature changes and phase transitions. Use the appropriate heat capacities and latent heats at each step.

Answer 1

The Correct Option is B

To determine the total entropy change for the process of ice at \( -5^\circ C \) heating to vapor at \( 110^\circ C \) under atmospheric pressure, we must consider the phase changes and temperature changes involved. Here's a step-by-step breakdown:

1. Heating Ice from \( -5^\circ C \) to \( 0^\circ C \):
   The heat capacity (\(C_{p,m}\)) is used to calculate the entropy change during temperature changes.
   \(\Delta S_1 = \int_{268 \, \text{K}}^{273 \, \text{K}} \frac{C_{p,m}}{T} \, dT\)
2. Melting of Ice at \( 0^\circ C \):
   The change of phase from solid to liquid involves latent heat of fusion (\(\Delta H_m \, \text{fusion}\)).
   \(\Delta S_2 = \frac{\Delta H_m \, \text{fusion}}{T_f}\) where \(T_f = 273 \, \text{K}\)
3. Heating Water from \( 0^\circ C \) to \( 100^\circ C \):
   Entropy change for the water as it is heated to boiling point.
   \(\Delta S_3 = \int_{273 \, \text{K}}^{373 \, \text{K}} \frac{C_{p,m}}{T} \, dT\)
4. Vaporization of Water at \( 100^\circ C \):
   Latent heat of vaporization (\(\Delta H_m \, \text{vaporisation}\)) is involved for liquid changing to gas.
   \(\Delta S_4 = \frac{\Delta H_m \, \text{vaporisation}}{T_b}\) where \(T_b = 373 \, \text{K}\)
5. Heating Vapor from \( 100^\circ C \) to \( 110^\circ C \):
   Further heating after vaporization.
   \(\Delta S_5 = \int_{373 \, \text{K}}^{383 \, \text{K}} \frac{C_{p,m}}{T} \, dT\)

The total entropy change (\(\Delta S_{\text{total}}\)) for the entire process is the sum of all individual changes:
\(\Delta S_{\text{total}} = \Delta S_1 + \Delta S_2 + \Delta S_3 + \Delta S_4 + \Delta S_5\)

Among the given options, the correct formulation for entropy change, considering phase transitions at their respective temperatures, is:
\(\Delta S_{\text{process}} = \int_{268 \, \text{K}}^{273 \, \text{K}} \frac{C_{p,m}}{T} \, dT + \frac{\Delta H_m \, \text{fusion}}{T_f} + \frac{\Delta H_m \, \text{vaporisation}}{T_b}\)