Calculate the entropy change accompanying conversion of 1 mole of ice at 273 K and 1 atm pressure into steam at 373 K and 1 atm pressure, given that at 273 K, the molar heat of fusion of ice is 6 kJ/mol and at 373 K, the molar heat of vapourization of water is 40.60 kJ/mol. Also assume that the molar heat capacity in the temperature range 373-273 K remains constant as 75.2 J/K per mol.
To calculate the entropy change, we need to consider three steps for this process:
1. Melting the ice at 273 K
2. Heating the water from 273 K to 373 K
3. Vaporizing water at 373 K
For each step, we'll calculate the entropy change individually and then add them up to find the total entropy change.
1. Melting the ice at 273 K:
The entropy change for melting ice can be calculated using the following formula:
ΔS_melting = ΔH_melting / T_melting
Where ΔH_melting is the molar heat of fusion at 273 K, which is given as 6 kJ/mol, and T_melting is the temperature, 273 K. Plugging in the values, we get:
ΔS_melting = (6 kJ/mol) / (273 K) = 0.02196 kJ/K per mol
2. Heating the water from 273 K to 373 K:
For this step, we can use the formula for the entropy change of a substance being heated at constant pressure:
ΔS_heating = C_p * ln(T2/T1)
Where C_p is the molar heat capacity, which is given as 75.2 J/K per mol, T1 is the initial temperature (273 K), and T2 is the final temperature (373 K).
First, calculate the natural logarithm of the ratio of the final and initial temperatures:
ln(T2/T1) = ln(373 K / 273 K) = 0.3127
Now, plug this value and the molar heat capacity into the formula for the entropy change during heating:
ΔS_heating = (75.2 J/K per mol)(0.3127) = 23.504 J/K per mol
However, since we need the answer in kJ/K, we will convert the units as follows:
ΔS_heating = 23.504 J/K per mol * (1 kJ / 1000 J) = 0.023504 kJ/K per mol
3. Vaporizing water at 373 K:
The entropy change for vaporizing water can be calculated using the following formula:
ΔS_vaporizing = ΔH_vaporizing / T_vaporizing
Where ΔH_vaporizing is the molar heat of vaporization at 373 K, which is given as 40.60 kJ/mol, and T_vaporizing is the temperature, 373 K. Plugging in the values, we get:
ΔS_vaporizing = (40.60 kJ/mol) / (373 K) = 0.10882 kJ/K per mol
Now, we can add up all the entropy changes for the three steps:
Total entropy change = ΔS_melting + ΔS_heating + ΔS_vaporizing
= 0.02196 kJ/K per mol + 0.023504 kJ/K per mol + 0.10882 kJ/K per mol
= 0.154284 kJ/K per mol
The total entropy change accompanying conversion of 1 mole of ice at 273 K and 1 atm pressure into steam at 373 K and 1 atm pressure is 0.154284 kJ/K per mol.
To calculate the entropy change accompanying the conversion of ice into steam, we need to consider the entropy changes at different steps of the process.
Step 1: Melting of ice
At 273 K and 1 atm pressure, the molar heat of fusion of ice (ΔH_fus) is given as 6 kJ/mol. The entropy change (ΔS_melt) during the melting process is related to ΔH_fus by the equation:
ΔS_melt = ΔH_fus / T_melt
where T_melt is the melting point temperature of ice (273 K).
Substituting the values, we find:
ΔS_melt = 6 kJ/mol / 273 K
Step 2: Raising the temperature of liquid water
The molar heat capacity of water in the temperature range from 273 K to 373 K (ΔCp) is given as 75.2 J/K per mol. The mass-specific heat capacity (c) can be calculated by dividing ΔCp by the molar mass of water (18 g/mol):
c = ΔCp / M
where M is the molar mass of water (18 g/mol).
Substituting the values, we find:
c = 75.2 J/K per mol / 18 g/mol
Next, we calculate the molar heat (ΔH_temp) required to raise the temperature of 1 mole of water from 273 K to 373 K using the equation:
ΔH_temp = c * ΔT
where ΔT is the change in temperature.
Substituting the values, we find:
ΔH_temp = c * (373 K - 273 K)
Step 3: Vaporization of liquid water
At 373 K and 1 atm pressure, the molar heat of vaporization of water (ΔH_vap) is given as 40.60 kJ/mol. The entropy change (ΔS_vap) during the vaporization process is related to ΔH_vap by the equation:
ΔS_vap = ΔH_vap / T_vap
where T_vap is the boiling point temperature of water (373 K).
Substituting the values, we find:
ΔS_vap = 40.60 kJ/mol / 373 K
Finally, the total entropy change (ΔS_total) is given by the sum of the entropy changes from each step:
ΔS_total = ΔS_melt + ΔS_temp + ΔS_vap
Substituting the given values and calculated quantities, we can compute the result.