Analysing Internal Energy and State Functions

Let's break down the two statements provided regarding internal energy and perfect gases.

Statement a: Internal Energy as a State Function

The statement claims that internal energy is a state function, depending only on the current state of the system and being independent of the path taken to reach that state.

• A state function (or function of state) is a property of a system that depends only on the current state of the system, not on the way the system acquired that state. Examples include temperature, pressure, volume, enthalpy, entropy, and internal energy.
• This means that if a system changes from an initial state to a final state, the change in a state function is always the same, regardless of the process (path) followed.

Based on the fundamental principles of thermodynamics, internal energy ($ U $) is indeed defined as a state function. It is determined by variables like temperature, pressure, and volume (or other relevant state variables), and its value changes only when the state changes, not because of the path taken (e.g., whether work or heat transfer dominated the process).

Therefore, statement (a) is true.

Statement b: Translational Internal Energy of a Perfect Gas

The statement claims that the translational contribution to the molar internal energy of a perfect gas is 3.72 kJ/mol at 25°C.

• For a perfect monatomic gas, the only contribution to internal energy comes from translational motion. According to the equipartition theorem, each translational degree of freedom contributes $ \frac{1}{2}kT $ per molecule, or $ \frac{1}{2}RT $ per mole, where $ k $ is the Boltzmann constant and $ R $ is the ideal gas constant.
• A perfect gas molecule has three translational degrees of freedom (motion along x, y, and z axes).
• So, the total translational internal energy per mole for a perfect gas is $ 3 \times \frac{1}{2}RT = \frac{3}{2}RT $.

Let's calculate this value at 25°C. The temperature must be in Kelvin:

$ T = 25^\circ\text{C} + 273.15 = 298.15 \text{ K} $

Using the ideal gas constant $ R = 8.314 \text{ J/mol·K} $:

\( \text{Translational energy} = \frac{3}{2} \times 8.314 \text{ J/mol·K} \times 298.15 \text{ K} \)

\( \text{Translational energy} = 1.5 \times 8.314 \times 298.15 \text{ J/mol} \)

\( \text{Translational energy} \approx 3723.4 \text{ J/mol} \)

Converting Joules to Kilojoules:

\( 3723.4 \text{ J/mol} = \frac{3723.4}{1000} \text{ kJ/mol} \approx 3.723 \text{ kJ/mol} \)

The calculated value of approximately 3.72 kJ/mol matches the value given in statement (b).

Therefore, statement (b) is also true.

Relationship Between Statements a and b

Both statements a and b are true. Now we consider if statement a is the correct explanation for statement b, as suggested by one of the options.

Statement (a) establishes that internal energy is a state function, meaning its value (including its components like translational energy) is determined solely by the current state (like temperature and pressure for a perfect gas) and not by the path. Statement (b) gives a specific value for the translational component of molar internal energy at a particular state (perfect gas at 25°C).

Because internal energy is a state function (a), the translational internal energy of a perfect gas at a defined state (like 25°C) must have a unique and definite value determined by that state. Statement (b) provides this specific value for the given state. The fact that internal energy is a state function underpins why we can calculate a specific value for it (or its components) at a given state using parameters like temperature and justifies that this value is characteristic of the state itself, regardless of how the gas reached 25°C.

Therefore, statement (a), by defining internal energy's fundamental nature as a state function, provides the thermodynamic basis for why a specific, path-independent value for its translational component exists at a given temperature (as stated in b). In this context, (a) can be considered the correct explanation for (b).

Conclusion

Based on the analysis, both statement (a) and statement (b) are true. Statement (a) explains that because internal energy is a state function, its value at any given state, such as the translational energy at 25°C for a perfect gas described in (b), is fixed and dependent only on that state.