10 kg of cold water at `2^(@)C` absorbs 50J of heat from the surrounding which were at a temperature of`37^(@)C` . What is the entropy change of (i) the system (ii) the surroundings (iii) the universe ? Assume the change in temperature of the water and the surroundings to be negligible.

Explanation

To solve the problem, we need to calculate the entropy change for the system (the cold water), the surroundings, and the universe. We will use the formula for entropy change, which is given by:

\[ \Delta S = \frac{Q}{T} \]

where \( \Delta S \) is the change in entropy, \( Q \) is the heat absorbed or released, and \( T \) is the absolute temperature in Kelvin.

### Step 1: Calculate the Entropy Change of the System (Cold Water)

1. **Identify the parameters**:
- Mass of water, \( m = 10 \, \text{kg} \)
- Heat absorbed, \( Q = 50 \, \text{J} \)
- Initial temperature of water, \( T_{water} = 2^\circ C = 275 \, \text{K} \) (convert to Kelvin by adding 273)

2. **Calculate the entropy change for the system**:
\[ \Delta S_{system} = \frac{Q}{T_{water}} = \frac{50 \, \text{J}}{275 \, \text{K}} \approx 0.1818 \, \text{J/K} \]

### Step 2: Calculate the Entropy Change of the Surroundings

1. **Identify the parameters**:
- Temperature of surroundings, \( T_{surroundings} = 37^\circ C = 310 \, \text{K} \)

2. **Since the surroundings lose heat, \( Q \) will be negative**:
\[ \Delta S_{surroundings} = \frac{-Q}{T_{surroundings}} = \frac{-50 \, \text{J}}{310 \, \text{K}} \approx -0.1613 \, \text{J/K} \]

### Step 3: Calculate the Entropy Change of the Universe

1. **The total entropy change of the universe is the sum of the entropy changes of the system and the surroundings**:
\[ \Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} \]
\[ \Delta S_{universe} = 0.1818 \, \text{J/K} - 0.1613 \, \text{J/K} \approx 0.0205 \, \text{J/K} \]

### Final Results

- (i) Entropy change of the system (cold water): \( \Delta S_{system} \approx 0.1818 \, \text{J/K} \)
- (ii) Entropy change of the surroundings: \( \Delta S_{surroundings} \approx -0.1613 \, \text{J/K} \)
- (iii) Entropy change of the universe: \( \Delta S_{universe} \approx 0.0205 \, \text{J/K} \)