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### Summary of the Problem:
The problem discusses air contamination inside a restaurant, specifically focusing on ventilation design to maintain appropriate air quality in different rooms. Two rooms are considered: one square and the other rectangular, and a balance of mass for the contaminants is required.

### Specific Tasks and Solutions:

1. **Write the mass balance equations for the other rooms.**
   - The provided equation is for one room (room 1), which is rectangular:
     \[
     \frac{dM_1}{dt} = \dot{m}_1 \cdot C_{A} + \dot{m}_{12} \cdot C_{B} - \dot{m}_2 \cdot C_{D} - \dot{m}_{vent_1} \cdot C_{E} 
     \]
     Here, \(\dot{m}_1\), \(\dot{m}_{12}\), etc., represent the mass flow rates and \(C_A\), \(C_B\), etc., represent the contaminant concentrations at various points.
   
   - For the square room (room 2), a similar mass balance equation can be written:
     \[
     \frac{dM_2}{dt} = \dot{m}_2 \cdot C_{F} + \dot{m}_{21} \cdot C_{G} - \dot{m}_1 \cdot C_{H} - \dot{m}_{vent_2} \cdot C_{I}
     \]
   
2. **Find the equilibrium concentrations for each room.**
   - At equilibrium, the change in mass of contaminants with respect to time is zero:
     \[
     \frac{dM_1}{dt} = 0 \quad \text{and} \quad \frac{dM_2}{dt} = 0
     \]
     This implies the inflow and outflow of contaminants in each room are balanced. The equilibrium concentrations can be found by solving the system of linear equations formed by setting the derivatives equal to zero in both mass balance equations.

3. **Determine the concentration of contaminants in the air flowing from room 2 to room 1 (\(C_{B}\)) using the equilibrium concentrations.**
   - Solve the mass balance equations for each room to find the equilibrium concentrations \(C_{A}\), \(C_{D}\), etc. Then, substitute these into the equation for \(C_{B}\).

4. **Analyze the case where there is no ventilation.**
   - If ventilation is removed (\(\dot{m}_{vent_1} = 0\) and \(\dot{m}_{vent_2} = 0\)), the equilibrium concentrations will be higher since contaminants accumulate in the rooms.

5. **Discuss how to optimize the ventilation system to minimize contaminant levels.**
   - The optimization could involve adjusting the mass flow rates \(\dot{m}_{vent_1}\) and \(\dot{m}_{vent_2}\) to ensure that contaminants are removed more efficiently.

### Further Questions:
1. How does increasing \(\dot{m}_{vent}\) affect the equilibrium concentration of contaminants?
2. What would happen if the air exchange between rooms (\(\dot{m}_{12}\) and \(\dot{m}_{21}\)) is increased?
3. How could the room sizes affect the ventilation requirements?
4. What are the potential health impacts if the contaminant levels exceed safe thresholds?
5. Could you model the contaminant concentrations over time instead of just at equilibrium?

### Tip:
When solving mass balance problems, it’s crucial to carefully account for all inflows and outflows, ensuring that units and concentrations are consistently applied across the equations.

Elabora balances de masa para las demás habitaciones y determina las concentraciones en equilibrio y la ventilación adecuada en un sistema de habitaciones interconectadas.

Analyze the mass balance in a system of interconnected rooms to determine equilibrium concentrations and optimal ventilation. This problem involves applying differential equations and mass conservation principles to maintain air quality in various rooms.

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