Animated Solution for Chemistry - Chemical Thermodynamics: The true statement amongst the following is

Imagine you are holding a cup of hot coffee. The heat slowly dissipates into the room, and the coffee cools down. This everyday phenomenon is governed by a fundamental concept in thermodynamics: entropy. Entropy, denoted by $S$, is often described as a measure of randomness or disorder in a system. But mathematically, it is much more precise.

When we heat a substance, we are transferring thermal energy into it. This energy causes the particles to move faster, vibrate more vigorously, and explore a larger number of microstates. Consequently, the randomness increases. This intuitive picture strongly suggests that entropy must be intimately connected to temperature.

To truly understand this connection, we must look at the formal definition of entropy. The infinitesimal change in entropy, $dS$, for a reversible process is defined as:

Here, $d{Q}_{\text{rev}}$ is the infinitesimal amount of heat exchanged reversibly, and $T$ is the absolute temperature at which the exchange occurs. Notice how $T$ is explicitly present in the denominator. Furthermore, the heat exchanged is itself related to the temperature change via the heat capacity ($C$) of the substance:

Substituting this back into our entropy equation, we get:

This equation is a clear mathematical declaration: the entropy $S$ is a function of temperature $T$. As temperature changes, entropy changes.

Entropy Change

Now, what if we want to find the total change in entropy, $\Delta S$, when a system goes from an initial temperature $T_1$ to a final temperature $T_2$? We simply integrate our infinitesimal expression:

Assuming the heat capacity $C$ is constant over this temperature range, the integration yields:

This result is profound. It tells us that the total entropy change $\Delta S$ is directly dependent on the initial and final temperatures. If you change the temperatures, you change the value of $\Delta S$. Therefore, $\Delta S$ is undeniably a function of temperature.

We have seen that both the absolute entropy $S$ (through its differential definition) and the entropy change $\Delta S$ (through integration) are mathematically tied to temperature. They are not static constants; they are dynamic properties that evolve as the thermal state of the system changes.

Therefore, the correct statement is that both $S$ and $\Delta S$ are functions of temperature. Understanding this dependence is crucial for mastering chemical thermodynamics and predicting the spontaneity of reactions!