A system goes from A to B via two processes I and II as shown in figure. If \Delta U_1 and \Delta U_

Q: What exactly is a state function?

A state function is a property whose value does not depend on the path taken to reach that specific value. It only depends on the current state of the system (like its pressure, volume, and temperature).

Q: Is the work done also the same for both processes?

No, work done is a path function. On a p-V diagram, work done is the area under the curve. Since Process I and Process II have different curves, the area under them (and thus the work done) will be different.

Q: If work is different, is the heat supplied the same?

No, heat is also a path function. According to the First Law of Thermodynamics, $\Delta Q = \Delta U + \Delta W$. Since $\Delta U$ is the same but $\Delta W$ is different, the heat supplied $\Delta Q$ must also be different for the two paths.

Q: What are some other examples of state functions?

Common state functions include Pressure ($p$), Volume ($V$), Temperature ($T$), Entropy ($S$), and Enthalpy ($H$).

Q: Why is internal energy considered a state function?

Internal energy is the sum of the microscopic kinetic and potential energies of the molecules. For an ideal gas, it depends only on temperature. Since temperature is a state variable, internal energy is also a state variable.

Question

A system goes from $A$ to $B$ via two processes I and II as shown in figure. If $\Delta U_1$ and $\Delta U_2$ are the changes in internal energies in the processes I and II respectively, then

Options:

(A) $\Delta U_1 = \Delta U_2$

(B) relation between $\Delta U_1$ and $\Delta U_2$ cannot be determined

(C) $\Delta U_2 > \Delta U_1$

(D) $\Delta U_2 < \Delta U_1$

Sigma Percentile · Accepted Answer

Imagine you are standing at the base of a mountain (State $A$) and you want to reach the summit (State $B$). You have two choices: you can take the steep, direct trail (Process I), or you can take the long, winding scenic route (Process II).

If I ask you, "How much distance did you walk?" your answer will heavily depend on the path you chose. The winding route will rack up far more miles than the direct trail. Distance, in this analogy, is a **path function**.

But what if I ask you, "How much did your altitude change?" It doesn't matter if you took the steep trail, the winding route, or even if you took a helicopter. Your change in altitude is simply the altitude at the summit minus the altitude at the base. Altitude is a **state function**.

### The Thermodynamic Equivalent

In thermodynamics, we deal with similar concepts. When a gas expands or compresses from an initial state $A$ to a final state $B$, it can do so via infinite possible paths on a $p-V$ diagram.

The work done by the gas ($W$) is the area under the $p-V$ curve. Just like the distance walked on the mountain, the area under the curve is different for Process I and Process II. Therefore, work is a **path function**.

Similarly, the heat supplied to the system ($Q$) also depends on the path taken.

### The Magic of Internal Energy

However, the internal energy ($U$) of an ideal gas depends solely on its temperature, which is uniquely defined by its pressure and volume at any given point ($pV = nRT$).

Because internal energy only cares about *where* the system is, and not *how* it got there, it is a **state function**.

For Process I, the change in internal energy is:
$$\Delta U_1 = U_B - U_A$$

For Process II, the change in internal energy is:
$$\Delta U_2 = U_B - U_A$$

Since both processes start at the exact same state $A$ and end at the exact same state $B$, the change in internal energy must be identical.

Therefore, we can confidently conclude:
**$$\Delta U_1 = \Delta U_2$$**

This simple yet profound realization is the cornerstone of the First Law of Thermodynamics!

Animated Solution for Physics - Thermodynamics: A system goes from $A$ to $B$ via two processes I and II as shown in figure. If $Δ U_{1}$ and $Δ U_{2}$ are the changes in internal energies in the processes I and II respectively, then

Select Answer:

Visualized Solution

Visualizing the Processes

Internal Energy as a State Function

Comparing the Two Processes

Final Conclusion

The Sigma Insight: First Law of Thermodynamics

The Thermodynamic Equivalent

The Magic of Internal Energy

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