What is the change in entropy when `2.5` mole of water is heated from `27^(@)C` to `87^(@)C`?
Assume that the heat capacity is constant `(C_(p))_(m)(H_(2)O)=4.2J//g=k,ln(1.2)=0.18)`

Options

A. 16.6 J/K
B. 9 J/K
C. 34.02 J/K
D. 1.89 J/K

Explanation

To find the change in entropy when 2.5 moles of water is heated from 27°C to 87°C, we can follow these steps:

### Step 1: Convert temperatures from Celsius to Kelvin
The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273 \]

For \( T_1 = 27°C \):
\[ T_1 = 27 + 273 = 300 \, K \]

For \( T_2 = 87°C \):
\[ T_2 = 87 + 273 = 360 \, K \]

### Step 2: Calculate the mass of water in grams
We know the number of moles of water and the molecular weight of water:
- Number of moles (\( n \)) = 2.5 moles
- Molecular weight of water (\( H_2O \)) = 18 g/mol

To find the mass (\( m \)):
\[ m = n \times \text{molecular weight} \]
\[ m = 2.5 \, \text{moles} \times 18 \, \text{g/gmol} = 45 \, g \]

### Step 3: Use the formula for change in entropy at constant pressure
The change in entropy (\( \Delta S \)) at constant pressure is given by:
\[ \Delta S = n C_p \ln \left( \frac{T_2}{T_1} \right) \]

Where:
- \( C_p \) is the heat capacity at constant pressure, given as \( 4.2 \, \text{J/g·K} \).
- \( n = 45 \, g \) (from step 2).
- \( T_1 = 300 \, K \) and \( T_2 = 360 \, K \) (from step 1).

### Step 4: Substitute the values into the entropy formula
Using the values we have:
\[ \Delta S = 45 \, g \times 4.2 \, \text{J/g·K} \times \ln \left( \frac{360}{300} \right) \]

### Step 5: Calculate \( \ln \left( \frac{360}{300} \right) \)
We know from the problem that:
\[ \ln(1.2) = 0.18 \]
Thus:
\[ \ln \left( \frac{360}{300} \right) = \ln(1.2) = 0.18 \]

### Step 6: Calculate the change in entropy
Now substituting back into the equation:
\[ \Delta S = 45 \, g \times 4.2 \, \text{J/g·K} \times 0.18 \]

Calculating:
\[ \Delta S = 45 \times 4.2 \times 0.18 \]
\[ \Delta S = 34.02 \, \text{J/K} \]

### Final Answer
The change in entropy when 2.5 moles of water is heated from 27°C to 87°C is:
\[ \Delta S = 34.02 \, \text{J/K} \]

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