91Ó°ÊÓ

Internal energy is a crucial concept in thermodynamics. It refers to the total energy contained within a system, which includes kinetic and potential energy at the molecular level. For an ideal gas, internal energy depends only on the temperature and the amount of gas. The formula to calculate internal energy is given as:
\( U = mc_vT \)
Here, \( m \) is the mass, \( c_v \) is the specific heat capacity at constant volume, and \( T \) is the temperature in Kelvin.
To solve the given problem, you first convert the temperatures from Celsius to Kelvin. Next, you use the internal energy formula to calculate the initial internal energies of both tanks. The final internal energy is the sum of these initial energies, as energy is conserved during the mixing process.

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as:
\( PV = nRT \)
Where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

In this problem, the volume remains constant during the mixing process. By knowing the final temperature, the total mass, and the specific gas constant \( R = 0.287 \text{ kJ/kg.K} \), you can rearrange the ideal gas law to find the final pressure.
\( P_f = \frac{m_fRT_f}{V_f} \ \)

Entropy is a measure of the randomness or disorder of a system. When two gases mix, the total entropy changes due to the heat exchange and mixing of molecules. The change in entropy for each tank can be calculated using the formula:
\( \Delta S = m c_v \ln( \frac{T_f}{T_{initial}}) \)

• \( \Delta S \) is the change in entropy
• \( m \) is the mass
• \( c_v \) is the specific heat capacity
• \( T_f \) is the final temperature
• \( T_{initial} \) is the initial temperature

By calculating the entropy changes for both tanks separately, you can determine the total entropy change of the system. In this specific problem, it is important to notice the calculated entropy change can initially appear negative, which is a point to carefully consider since entropy in a closed system should increase or remain constant.

Determining the final temperature after the mixing of two gases involves the application of the conservation of energy principle. The total internal energy of the system remains constant.
The final temperature \( T_f \) can be calculated using the equation:

\( T_f = \frac{U_f}{m_f c_v} \)

• \( U_f \) is the final internal energy
• \( m_f \) is the total mass
• \( c_v \) is the specific heat capacity

In this problem, by combining the initial internal energies and masses, you can find the value for the final temperature after the system reaches equilibrium.

The final pressure of the mixed gases can be found using the ideal gas law, considering that the volume remains constant. Once you have the final temperature and the total mass, you can calculate the final pressure using:
\( P_f = \frac{m_fRT_f}{V_f} \)
Since the volume doesn't change, you can use the total moles of gas (considering the combined mass and specific gas constant) along with the computed final temperature to find the final pressure. This approach ensures you understand how all quantities are interconnected in a thermodynamic system.