Let's analyze the given assertion and reason regarding heat, work, and their relationship with thermodynamic states.

The question states an assertion that heat and work transfer cannot be expressed as the difference between end states, and a reason stating that heat and work are both exact differentials.

Understanding Heat and Work in Thermodynamics

In thermodynamics, heat and work are forms of energy transfer that occur during a process as a system changes from one state to another. They are not properties of the system itself, but rather describe the interactions that happen across the system boundary during a transition.

Assertion (A): Heat and work transfer cannot be expressed as difference between the end states.

This statement is about whether heat and work are 'state functions' or 'path functions'.

• A state function (or property) depends only on the current state of the system, not on the path taken to reach that state. Examples include internal energy (\$\$U\$\$), enthalpy (\$\$H\$\$), temperature (\$\$T\$\$), pressure (\$\$P\$\$), and volume (\$\$V\$\$). The change in a state function between two states is simply the difference between its values at the final and initial states (\$\$dU = U_{final} - U_{initial}\$\$).
• A path function depends on the specific process or path taken as the system moves between two states. Heat transfer (\$\$Q\$\$) and work transfer (\$\$W\$\$) are classic examples of path functions. The amount of heat added or work done depends on the way the process is carried out, not just the initial and final states.

Since heat and work are path functions, their values depend on the path taken, not just the end states. Therefore, the change in heat or work cannot be simply calculated as the difference between their 'values' at the end states, because they don't have defined values at a state; they only exist as energy transfer during a process.

Hence, Assertion (A) is true.

Reason (R): Heat and work are both exact differentials.

This statement is about the mathematical nature of the infinitesimals representing small amounts of heat and work transfer.

• An exact differential corresponds to a state function. If \$\$df\$\$ is an exact differential, its integral between two points A and B is path-independent: \$\$\int_A^B df = f_B - f_A\$\$. State functions like internal energy (\$\$U\$\$) have exact differentials (\$\$dU\$\$).
• An inexact differential corresponds to a path function. If \$\$\delta g\$\$ is an inexact differential, its integral between two points A and B depends on the path taken: \$\$\int_A^B \delta g\$\$ is path-dependent. Heat (\$\$\delta Q\$\$) and work (\$\$\delta W\$\$) are inexact differentials. They are typically denoted by \$\$\delta\$\$ instead of \$\$d\$\$ to indicate this property.

Since heat and work are path functions, their infinitesimal changes (\$\$\delta Q\$\$ and \$\$\delta W\$\$) are inexact differentials. The integral of an inexact differential between two states depends on the path of integration.

Therefore, Reason (R), which states that heat and work are both exact differentials, is false.

Conclusion

Based on the analysis:

• Assertion (A) is true because heat and work are path functions and their transfer depends on the process path, not just the end states.
• Reason (R) is false because heat and work are inexact differentials, not exact differentials.

Thus, Assertion (A) is true but Reason (R) is false.