Question

Question: One mole of a monatomic ideal gas begins in a state with $\mathit{P}\mathbf{=}\mathbf{1}\mathbf{.00}\mathbf{}\mathbf{\text{atm}}$and$\mathit{T}\mathbf{=}\mathbf{300}\mathbf{}\mathbf{\text{K}}$. It is expanded reversibly and adiabatically until the volume has doubled; then it is expanded irreversibly and isothermally into a vacuum until the volume has doubled again; and then it is heated reversibly at constant volume to 400K. Finally, it is compressed reversibly and isothermally until a final state with$\mathit{P}\mathbf{=}\mathbf{1}\mathbf{.00}\mathbf{}\mathbf{\text{atm}}$and$\mathit{T}\mathbf{=}\mathbf{400}\mathbf{}\mathbf{\text{K}}$is reached. Calculate$\mathit{\Delta }{\mathit{S}}_{\mathbf{\text{sys}}}$for this process.

Short answer

The overall enthalpy change$\Delta S_{\text{sys}}$ is $5.98\text{J}{\text{K}}^{-1}$.

Step-by-step solution

Given data

Initial temperature $T_i=300\text{K}$, Final temperature $T_f=400\text{K}$, amount of monatomic gas$n=1.00\text{mol}$ and Pressure $p_1=1.00\text{atm}$.

Concept used

Entropy Change is the phenomenon that is the measure of change of disorder or randomness in a thermodynamic system. It is related to the conversion of heat or enthalpy in work. A thermodynamic system that has more randomness means it has high entropy.

Calculation of entropy change

Entropy is a feature of the state of a thermodynamic system. This means that it does not depend on the path of the process, but only on the initial and final state.

So, there is no need to calculate the change in entropy for each individual process.

The entropy change formula for an adiabatic process is as follows:

$\Delta S=n{C}_p\ln \frac{T_f}{T_i}$ ........ (1)

Where$C_p$ is the heat capacity, whose value is$\frac{5}{2}R$ for monatomic gases.

$\begin{array}{l}n=1.00\text{mol}\\ T_i=300\text{K}\\ T_f=400\text{K}\end{array}$

Substitute these values in equation (1)

$\begin{array}{c}\Delta S=1.00\times \frac{5}{2}\times 8.314\times \ln \frac{400}{300}\\ \Delta S=5.98\text{J}{\text{K}}^{-1}\end{array}$