Question

A hot body with constant heat capacity 800 //K at temperature 925 K is dropped gently into a vessel containing 1 kg of water at temperature 300 K and the combined system is allowed to reach equilibrium. The change in the total entropy $\bigtriangleup S$ is _____________ J/K (Round off to 1 decimal place).
[Take the specific heat capacity of water to be 4200 J/kg K. Neglect any loss of heat to the vessel and air and change in the volume of water.]

Answer 1

Correct Answer: 537.5

To compute the change in the total entropy \( \Delta S \) of the system, we consider both the hot body and the water. The heat exchanges between them can be calculated using their heat capacities and the principle of energy conservation. The steps are as follows:

1. **Define the initial conditions and variables:**
The heat capacity of the hot body, \( C_h = 800 \) J/K.
The initial temperature of the hot body, \( T_h = 925 \) K.
The mass of the water, \( m_w = 1 \) kg.
The specific heat capacity of water, \( c_w = 4200 \) J/kg·K.
The initial temperature of the water, \( T_w = 300 \) K.

2. **Calculate the final equilibrium temperature \((T_f)\):**
At thermal equilibrium, the heat lost by the hot body equals the heat gained by the water:
\[ C_h(T_h - T_f) = m_wc_w(T_f - T_w) \]
Substitute the known values:
\[ 800(925 - T_f) = 4200(1)(T_f - 300) \]
Solve this equation for \( T_f \).
After simplifying:
\[ 740000 - 800T_f = 4200T_f - 1260000 \]
Combine terms:
\[ 2000T_f = 2000000 \]
\[ T_f = 625 \text{ K} \]

3. **Calculate the entropy change of the hot body \((\Delta S_h)\):**
The entropy change is given by:
\[ \Delta S_h = C_h \ln\left(\frac{T_f}{T_h}\right) \]
\[ \Delta S_h = 800 \ln\left(\frac{625}{925}\right) \approx -322.9 \text{ J/K} \]

4. **Calculate the entropy change of the water \((\Delta S_w)\):**
The entropy change for the water is:
\[ \Delta S_w = m_wc_w \ln\left(\frac{T_f}{T_w}\right) \]
\[ \Delta S_w = 4200 \ln\left(\frac{625}{300}\right) \approx 860.4 \text{ J/K} \]

5. **Calculate the total entropy change \((\Delta S)\):**
\[ \Delta S = \Delta S_h + \Delta S_w \approx -322.9 + 860.4 = 537.5 \text{ J/K} \]
This is the change in the total entropy. 
The change in total entropy for the system is 537.5 J/K.