Question

Mixing entropy arises due to

• A. Volume change
• B. Heat transfer
• C. Work done
• D. Random distribution of components

Hint:

Spontaneous mixing of ideal substances involves no heat exchange or volume changes. The increase in entropy is purely a configurational entropy increase, which means it is caused entirely by the chaotic, random spatial distribution of the different molecules intermingling.

Answer 1

The Correct Option is D

Concept: The entropy of mixing (\(\Delta S_{\text{mix}}\)) is the change in total entropy that occurs when two or more distinct chemical substances are combined to form a uniform solution or gas mixture. From a statistical perspective, entropy evaluates structural disorder and spatial uncertainty. When different components are mixed, the individual molecules intermingle, creating a highly disordered arrangement compared to their separated, unmixed states. Statistical Mechanics Analysis:
Consider two containers containing distinct ideal gas species, \(A\) and \(B\), at identical pressures and temperatures.
• Before mixing, the molecules of species \(A\) are confined to their specific volume section, and species \(B\) molecules are confined to theirs. Each system has a relatively low number of spatial configurations.
• When the partition separating them is removed, the fluids mix spontaneously without any chemical reactions. The molecules of both species can now occupy the entire combined volume.
• This creates a massive increase in the total number of available microscopic spatial configurations (\(\Omega\)) for the mixture. The molecules are now distributed randomly throughout the space. According to Boltzmann's law (\(S = k_B \ln \Omega\)), this increase in structural configuration options leads directly to an increase in entropy. This change is given by the ideal mixing expression: \[ \Delta S_{\text{mix}} = -R \sum n_i \ln x_i \] where \(x_i\) is the mole fraction of each component. Because \(x_i < 1\), the term \(\ln x_i\) is negative, making \(\Delta S_{\text{mix}}\) always positive for spontaneous mixing. Let us evaluate why other options are incorrect for an ideal mixing process:
• Option (1) is incorrect: For ideal gases mixed at constant total pressure, the combined volume equals the sum of the individual volumes (\(\Delta V_{\text{mix}} = 0\)). The mixing entropy increase is driven by the distribution of the components rather than a change in overall system volume.
• Options (2) and (3) are incorrect: Ideal mixing is defined as an isothermal and adiabatic process where no heat is exchanged (\(\Delta Q = 0\)) and no boundary work is performed (\(\Delta W = 0\)). The entropy change is driven entirely by configuration entropy changes from the random distribution of components. Hence, option (4) is correct.