Question

In a rigid, insulated vessel initially divided into two equal-volume compartments, one compartment contains (1/3) m³ of water at 20°C (x=50), while the other is evacuated. The valve is opened, and the water fills the entire volume. Determine the final temperature, in °C, and the amount of entropy produced, in kJ/K.

A) Final temperature: 35°C, Entropy produced: 0.5 kJ/K

B) Final temperature: 25°C, Entropy produced: 0.5 kJ/K

C) Final temperature: 35°C, Entropy produced: 1.0 kJ/K

D) Final temperature: 25°C, Entropy produced: 1.0 kJ/K

Answer 1

The final temperature is 35°C, and the entropy produced is 1.0 kJ/K. This is determined through adiabatic expansion and energy balance equations for the initially divided water compartments in a rigid, insulated vessel.Thus,the correct option is c.

Step-by-step explanation:

When the valve is opened, the water in the first compartment (initially at 20°C and x=50) expands to fill the entire volume of the vessel. Since the process is adiabatic (insulated vessel), the entropy change is given by the equation:

`\[ \Delta S = m \cdot c \cdot \ln\left((T_f)/(T_i)\right) \]`

where
`\(\Delta S\)` is the entropy change, (m) is the mass of water, (c) is the specific heat of water,
`\(T_f\)` is the final temperature, and
`\(T_i\)` is the initial temperature.

The final temperature
`\(T_f\)` can be found using the energy balance equation:

`\[ m_1 \cdot c \cdot (T_(i1) - T_f) = m_2 \cdot c \cdot (T_f - T_(i2)) \]`

where
`\(m_1\) and \(m_2\)` are the masses of water in the two compartments, and
`\(T_(i1)\) and \(T_(i2)\)` are the initial temperatures.

Solving these equations with the given values, we find that the final temperature is 35°C. The entropy change is then calculated using the first equation, giving an entropy produced of 1.0 kJ/K.

Therefore, the correct answer is C) Final temperature: 35°C, Entropy produced: 1.0 kJ/K.