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The Ideal Gas Law is a fundamental equation in thermodynamics used to describe the behavior of gases. It combines various gas laws, like Boyle's, Charles's, and Avogadro's laws, into one formula. The equation is given by:
\[ PV = nRT \]
Here,

• *P* is the pressure of the gas,
• *V* is the volume,
• *n* is the number of moles,
• *R* is the universal gas constant,
• *T* is the temperature in absolute units (Kelvin or Rankine).

In our exercise, we employed the Ideal Gas Law to determine the initial temperature (*T鈧�*) of the carbon monoxide gas. Given that we had one lbmol of gas initially at 5 atm in a 90 ft鲁 vessel, we calculated *T鈧�* to be approximately 615.96 掳Rankine. This was a crucial step because it set the stage for all subsequent calculations.

Internal energy is a key concept in thermodynamics, representing the total energy contained within a system due to the microscopic motions of its particles. For an ideal gas, the change in internal energy (\[ \Delta U \] ). is given by: \[ \Delta U = n \cdot C_v \cdot \Delta T \] where

• *n* is the number of moles,
• *C_v* is the specific heat capacity at constant volume,
• \( \Delta T \) is the change in temperature.

Since the vessel in our problem was rigid (no volume change) and insulated (no heat transfer), the change in internal energy directly matched the electrical work done on the system, which was 1200 Btu.

In our exercise, using the specific heat capacity for carbon monoxide (*C_v* = 5 Btu/(lbmol鈰吢癛)), we determined the temperature change (\( \Delta T \) = 240 掳R). This allowed us to find the final temperature ( T鈧�) of the gas.

Electrical work involves transferring energy to a system using electrical power, which can result in changes in temperature and internal energy of the system. The formula utilized is:

\[ W = P \cdot t \] Here,

• *W* is the work done, usually measured in energy units like Btu,
• *P* is the power rate (Btu/s),
• *t* is the duration for which power is supplied (seconds).

In the problem, the electric resistor supplied energy at a constant rate of 10 Btu/s for a duration of 2 minutes (120 seconds). Hence, the total electrical work done was:
\[ W = 10 \ Btu/s \cdot 120 \ s = 1200 \ Btu \]
This electrical work was pivotal as it equated to the change in internal energy, given the conditions of the rigid and insulated vessel.

Exergy destruction quantifies the loss of system efficiency due to irreversibilities, such as friction or unresisted heat flow, within a thermodynamic process. It's an essential measure because it indicates the potential for improvement in energy conversion processes. The formula for exergy destruction is:
\[ \Delta Ex_d = W - \Delta Ex \] where

• *W* is the electrical work done on the system,
• \( \Delta Ex \) is the change in exergy.

In our specific exercise, exergy destruction was calculated by first finding the change in exergy ( \Delta Ex). This involves subtracting the product of the ambient temperature ( T鈧�) and change in entropy (\( \Delta S \)) from the change in internal energy. Finally, we determined the exergy destruction to be 885.55 Btu, highlighting significant irreversibilities during this energy transfer process.