Chemistry 351

Activity: Correcting for Non-Ideal Behavior

Throughout this chapter we have treated solutions as though they behaved ideally. In an ideal solution, the chemical potential of a component can be written as

\[ \mu_i=\mu_i^*+RT\ln x_i \]

where \(x_i\) is the mole fraction of component \(i\).

This model works well for very dilute solutions, but real solutions are rarely perfectly ideal. Molecules and ions interact with one another through intermolecular forces, hydrogen bonding, and electrostatic attractions. As a result, the behavior of a real solute is not always determined solely by its concentration.

For example, two solutions may contain the same concentration of ions, yet exhibit different thermodynamic behavior because the ions interact differently with their surroundings.

To account for these deviations from ideality, thermodynamics introduces the concept of activity, denoted by \(a\).

\[ \mu_i=\mu_i^*+RT\ln a_i \]

This expression has exactly the same form as the ideal-solution equation, but the activity replaces the concentration variable.

Activity may be viewed as an effective concentration—the concentration that produces the observed chemical potential.

Activity Coefficients

The activity of a solute is related to its concentration through the activity coefficient, \(\gamma\):

\[ a=\gamma\frac{m}{m^\circ} \]

where \(m\) is the molality of the solute and \(m^\circ\) is the standard molality (1 mol kg\(^{-1}\)).

The activity coefficient measures how strongly the solution deviates from ideal behavior.

• If \(\gamma = 1\), the solution behaves ideally.
• If \(\gamma \ne 1\), the solution exhibits non-ideal behavior.

As the concentration of a solute approaches zero,

\[ \gamma \rightarrow 1 \]

so all sufficiently dilute solutions approach ideal behavior. This is why Raoult's Law and Henry's Law work best in the dilute limit.

Why Activity Matters

Activity allows the simple thermodynamic equations developed for ideal solutions to be used for real solutions. Instead of changing the form of the chemical-potential equation, we simply replace concentration with activity.

This idea becomes especially important for electrolyte solutions. Positive and negative ions attract one another strongly, causing substantial deviations from ideal behavior. As a result, concentrations alone are often insufficient to describe the thermodynamics of ionic solutions.

In the next section, we will examine how activity coefficients for ionic solutes can be estimated using the Debye-Hückel theory.

Big picture: Concentration tells us how much solute is present, but activity determines the chemical potential. Activity therefore serves as the bridge between real solutions and the ideal models developed throughout thermodynamics, allowing the same equations to remain useful even when intermolecular interactions become important.

Practice

Question 1
Which statement best describes the purpose of activity? A. Activity measures the concentration of a solute in mol L\(^{-1}\). B. Activity is an effective concentration that accounts for non-ideal behavior. C. Activity measures the rate at which a solute dissolves. D. Activity is another name for mole fraction.
Show Answer Answer: B Activity is an effective concentration that reflects how a solute actually contributes to the chemical potential. It allows ideal-solution equations to be applied to real solutions.

Question 2
A solution becomes increasingly dilute. What happens to the activity coefficient, \(\gamma\)? A. It always approaches zero. B. It always becomes very large. C. It approaches one. D. It becomes equal to the molality.
Show Answer Answer: C As a solution approaches infinite dilution, intermolecular interactions become less important and the solution approaches ideal behavior. Consequently, \[ \gamma \rightarrow 1 \] in the dilute limit.

Question 3
The activity of a solute is often written as \[ a=\gamma\frac{m}{m^\circ} \] If the activity coefficient \(\gamma\) is dimensionless and \(m^\circ\) has the same units as \(m\), what are the units of the activity \(a\)? A. mol kg\(^{-1}\) B. kg mol\(^{-1}\) C. Activity has no units. D. The units depend on the value of \(\gamma\).
Show Answer Answer: C The ratio \[ \frac{m}{m^\circ} \] is dimensionless because the numerator and denominator have identical units. Since \(\gamma\) is also dimensionless, the activity is dimensionless as well. This is important because activity appears inside a logarithm in expressions such as \[ \mu=\mu^*+RT\ln a \] and logarithms require dimensionless arguments.

Key points (one glance)

• Ideal-solution models assume that the chemical potential depends only on concentration: \[ \mu_i=\mu_i^*+RT\ln x_i \]
• Real solutions often deviate from ideal behavior because molecules and ions interact with one another through intermolecular forces and electrostatic attractions.
• To account for these deviations, concentration is replaced by activity: \[ \mu_i=\mu_i^*+RT\ln a_i \]
• Activity may be interpreted as an effective concentration that determines the chemical potential of a species.
• Activity and concentration are related through the activity coefficient: \[ a=\gamma\frac{m}{m^\circ} \]
• The activity coefficient, \(\gamma\), measures the degree of non-ideal behavior.
• For an ideal solution: \[ \gamma=1 \] and the activity is equal to the normalized concentration.
• As a solution becomes increasingly dilute, \[ \gamma \rightarrow 1 \] so all sufficiently dilute solutions approach ideal behavior.
• Activities and activity coefficients are dimensionless quantities.
• Activity is particularly important for electrolyte solutions, where ionic interactions can produce substantial deviations from ideal behavior.

Big picture: Concentration tells us how much solute is present, but activity determines the chemical potential. By replacing concentration with activity, the simple thermodynamic equations developed for ideal solutions can be extended to real solutions, providing the foundation for the treatment of electrolyte solutions, chemical equilibrium, and electrochemistry.