Question

Four moles of an ideal gas are caused to expand from a volume \(V_{1}\) to a volume \(V_{2}=3.45 V_{1} .(a)\) If the expansion is isothermal at the temperature \(T=410 \mathrm{~K}\), find the work done on the expanding gas. ( \(b\) ) Find the change in entropy, if any. ( \(c\) ) If the expansion is reversibly adiabatic instead of isothermal, what is the entropy change?

Short answer

The work done on the gas during the isothermal expansion is approximately 11360 J. The entropy change during the isothermal expansion is around 8.2 J/K. In a reversible adiabatic expansion, the entropy change is zero.

Step-by-step solution

Calculate the Work done in Isothermal Expansion

The work done \(W\) on the gas during an isothermal expansion can be calculated using the formula \(W = nRT \ln{(V_2/V_1)}\). Here, \(n = 4\) moles, \(R = 8.3145\) J/K*mol is the gas constant, \(T = 410\) K is the temperature and \(V_2/V_1 = 3.45\). Substituting these values in the formula, we get \(W = 4 * 8.3145 * 410 * ln(3.45)\) J.

Find the Change in Entropy during the Isothermal Expansion

The entropy change \(\Delta S\) during the isothermal expansion can be calculated using the formula \(\Delta S = nR \ln{(V_2/V_1)}\). Here again, \(n = 4\) moles, \(R = 8.3145\) J/K*mol is the gas constant and \(V_2/V_1 = 3.45\). Substituting these values in the formula, we get \(\Delta S = 4 * 8.3145 * ln(3.45)\) J/K.

Entropy change during Reversible Adiabatic Expansion

In a reversible adiabatic process where the system is insulated, there is no heat exchange with the surroundings. Hence, the entropy change is zero.

Key concepts

Isothermal Expansion

In the context of thermodynamics, isothermal expansion is a process wherein a gas changes its volume while maintaining a constant temperature. Isothermal means 'equal temperature', signaling that throughout the expansion, the gas's temperature does not change. This constant temperature condition allows using the ideal gas law \(PV=nRT\), where \(P\) stands for pressure, \(V\) denotes volume, \(n\) the number of moles, \(R\) the gas constant, and \(T\) the absolute temperature.

An essential feature of isothermal expansion is that the gas does the work on the surroundings since the volume increases. The work \(W\) done by the gas can be computed by integrating the pressure over the change in volume. As the temperature remains constant, the work done during an isothermal process for an ideal gas is given by \(W = nRT \ln{(V_2/V_1)}\). Thus, understanding how to calculate this value is crucial for solving problems involving isothermal processes, such as the expansion of an ideal gas at constant temperature described in the exercise.

Entropy Change

Entropy is a measure of the disorder or randomness in a system and reflects the number of ways energy can be distributed within that system. In thermodynamics, the concept of entropy allows us to quantify the degree of irrevocability in a given process. During an isothermal expansion, such as the one described in the problem, the entropy change \(\Delta S\) of the system can be determined by the equation \(\Delta S = nR \ln{(V_2/V_1)}\).

This equation implies that when a gas expands isothermally, its volume increases, and so does its entropy, representing an increase in the number of possible microscopic configurations. Entropy change in any process provides insights into the energy dispersal within a system. In simpler terms, if a gas expands into a larger volume without changing temperature, it has more ways to arrange itself, and thus, the entropy increases. The calculation of the entropy change enhances understanding of the second law of thermodynamics and the overall direction of natural processes.

Reversible Adiabatic Process

A reversible adiabatic process is characterized by the absence of heat exchange between a system and its surroundings. When a process is adiabatic, it means that the system is perfectly insulated, so no thermal energy enters or leaves the system as work is done on or by the system. 'Reversible' indicates that the process can be reversed by an infinitesimally small change in the external conditions, without leaving any net change in the system and surroundings.

During such a process, any change in the internal energy of the system is equal to the work done by or on the system. The entropy change in a perfectly reversible adiabatic process is zero, as there is no heat transfer, which would otherwise contribute to a change in entropy. This concept deeply intertwines with the first and second laws of thermodynamics, suggesting that reversible adiabatic processes are idealizations used to simplify the analysis of thermodynamic systems. Understanding reversible adiabatic expansion is pivotal for comprehending more complex thermodynamic cycles and processes.