Question

An ideal gas undergoes a reversible isothermal expansion at ${132}^{\circ }\mathrm{C}$. The entropy of the gas increases by $46.2\mathrm{J}/\mathrm{K}$. How much heat is absorbed?

Short answer

The heat absorbed by the gas is approximately 18711 J.

Step-by-step solution

Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin. The formula to do this is K = C + 273. So, $T={132}^{\circ }C+273=405K$

Understand the Given Values

We know that the entropy change $\Delta S=46.2J/K$. The heat absorbed or expelled in a process is given by $Q=T\Delta S$. In this case, because we’re dealing with heat absorption and the entropy increase $\Delta S>0$, we know that $Q>0$.

Calculate the Heat Absorbed

Substitute the values of T and ΔS into the equation. This gives us $Q=T\Delta S=405K\ast 46.2J/K=18711J$.

Key concepts

Reversible Isothermal Expansion

In the realm of thermodynamics, reversible isothermal expansion is a fascinating concept that involves the expansion of an ideal gas at a constant temperature. For this process to take place isothermally, the system must be in thermal equilibrium with its surroundings, maintaining a constant temperature throughout the expansion. This means that any heat added to the system is immediately used to do work, ensuring that temperature remains unchanged. This process is termed 'reversible' because it can be reversed by infinitesimal changes in external conditions, without any net energy change in the universe. This ideal condition is important because it provides a benchmark for the most efficient possible process, where energy losses are minimized.

Entropy Change

Entropy, in simple terms, is a measure of disorder or randomness within a system. During the reversible isothermal expansion, the entropy of the system changes because the gas particles spread out and occupy a larger volume at the same temperature.In thermodynamics, the change in entropy ($\Delta S$) is an important parameter because it helps predict the direction of thermal energy flow. For the given exercise, we see an entropy change of $46.2J/K$, indicating an increase in the system's disorder. Entropy change is often associated with the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time. Hence, as the gas expands, the increase in entropy suggests that the process is spontaneous.

Ideal Gas Law

The Ideal Gas Law is an essential equation in understanding the behavior of gases under various conditions. It is expressed as $PV=nRT$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the gas constant, and $T$ is temperature.This equation aptly describes the relationship between these variables for an ideal gas, which is a hypothetical gas that perfectly follows the kinetic molecular theory. In our context of isothermal expansion, the temperature $T$ remains constant, which implies that any increase in volume $V$ must be accompanied by a corresponding decrease in pressure $P$ to maintain the equality.The Ideal Gas Law thus provides a vital link in calculating various properties of gases when considering entropy changes or heat absorption.

Heat Absorption

In a reversible isothermal expansion, heat absorption is a key aspect because it balances the increase in internal energy needed to perform work against the surroundings at a constant temperature.To calculate the amount of heat absorbed, one can use the relationship $Q=T\Delta S$. Here, $T$ represents the temperature in Kelvin, and $\Delta S$ denotes the change in entropy. For the provided example, substituting $T=405K$ and $\Delta S=46.2J/K$ gives us the heat absorbed, $Q=18711J$. This equation highlights the direct proportionality between heat and entropy in such scenarios, resonating with the thermodynamic paradigm at work.This calculation not only illustrates the linkage between heat energy and entropy changes but also emphasizes the efficiency of an isothermal process, where energy is carefully balanced between the system and its surroundings.