Activities & Activity Coefficients - Physical Chemistry 1

1. In a solution containing 0.05 M Na₂SO₄, the mean ionic activity coefficient is found to be 0.544. If an additional 0.02 M NaCl is added to this solution (assume no volume change), and the new mean ionic activity coefficient for Na₂SO₄ becomes 0.445 due to increased ionic strength, what is the percent change in the activity of Na₂SO₄?
   Answer: The activity decreases by 18.2% due to the salt effect from added NaCl When you encounter questions about ionic activities in mixed electrolyte solutions, focus on how ionic strength affects activity coefficients and remember that activity depends on both concentration and the activity coefficient. To find the percent change in Na₂SO₄ activity, you need to calculate the activity before and after NaCl addition. For Na₂SO₄, the mean ionic activity is $a_± = (C_{Na^+})^2 \cdot C_{SO_4^{2-}} \cdot (\gamma_±)^3$, but this simplifies to $a = C \cdot \gamma_±^3$ when comparing the same electrolyte at the same concentration. Initially: $a_1 = 0.05 \times (0.544)^3 = 0.05 \times 0.161 = 0.00805$ After adding NaCl: $a_2 = 0.05 \times (0.445)^3 = 0.05 \times 0.0881 = 0.00441$ Percent change = $\frac{0.00441 - 0.00805}{0.00805} \times 100\% = -18.2\%$ Choice A correctly identifies this 18.2% decrease and properly attributes it to the salt effect—when NaCl increases the ionic strength, it lowers the activity coefficient of Na₂SO₄ through enhanced electrostatic interactions. Choice B gives an incorrect percentage (22.5%), likely from miscalculating the cubic relationship. Choice C incorrectly assumes activity equals concentration, ignoring the activity coefficient's role entirely. Choice D not only has the wrong sign but also misunderstands that increased ionic strength typically decreases activity coefficients in dilute solutions. Remember: Activity changes can occur even when concentration stays constant if the activity coefficient changes due to ionic strength effects from other electrolytes—this is the classic "salt effect."
2. For a binary electrolyte solution following the Debye-Hückel limiting law, the activity coefficient γ± of the electrolyte decreases as ionic strength increases. If the mean ionic activity coefficient of CaCl₂ in a 0.010 M solution is 0.850, and in a 0.040 M solution is 0.718, what can be concluded about the relationship between activity and concentration for this electrolyte over this concentration range?
   Answer: Activity increases with concentration but at a decreasing rate due to the competing effects of increasing concentration and decreasing activity coefficient When you encounter questions about electrolyte solutions and activity coefficients, remember that activity combines both concentration effects and deviations from ideal behavior. The key relationship is: activity = concentration × activity coefficient. Let's calculate the mean ionic activities for CaCl₂ at both concentrations. For CaCl₂, which dissociates into 3 ions (Ca²⁺ + 2Cl⁻), we need the mean ionic concentration: $c_± = c \times (z_+^{z_+} \times z_-^{z_-})^{1/(z_+ + z_-)} = c \times (1^1 \times 2^2)^{1/3} = c \times 4^{1/3} = 1.587c$ At 0.010 M: $a_± = γ_± \times c_± = 0.850 \times (1.587 \times 0.010) = 0.0135$ At 0.040 M: $a_± = γ_± \times c_± = 0.718 \times (1.587 \times 0.040) = 0.0455$ The activity clearly increases from 0.0135 to 0.0455, but notice the rate of increase is slowing. While concentration quadrupled, activity only increased by a factor of 3.37 due to the decreasing activity coefficient. Answer A is wrong because the relationship isn't linear—the activity coefficient doesn't fully compensate for non-ideality. Answer C is incorrect since activity clearly changes rather than remaining constant. Answer D is wrong because activity increases, not decreases—the concentration effect outweighs the activity coefficient decrease. Answer B correctly identifies that activity increases with concentration but at a decreasing rate due to these competing effects. Study tip: Always calculate the actual activities when given both concentrations and activity coefficients—this reveals the true behavior pattern rather than just theoretical trends.
3. A solution of ethanol and water at 25°C shows negative deviation from Raoult's law. If the activity of ethanol is 0.25 when its mole fraction is 0.40, what does this reveal about the molecular interactions?
   Answer: Ethanol-water hydrogen bonds are stronger than pure component interactions, stabilizing the solution Negative deviation (activity < mole fraction, since γ = a/x = 0.25/0.40 = 0.625 < 1) indicates that intermolecular attractions between unlike molecules (ethanol-water) are stronger than those between like molecules. This stabilizes the solution and reduces the escaping tendency of each component. Choice A incorrectly describes weaker interactions. Choice C misattributes the effect to excluded volume. Choice D ignores the clear non-ideal behavior.
4. A student measures the activity coefficient of HCl at different concentrations and finds that the values deviate from Debye-Hückel limiting law predictions at concentrations above 0.01 M. The experimental data shows γ± = 0.875 at 0.02 M versus the predicted γ± = 0.823 from limiting law. What is the most likely explanation for this discrepancy, and what does it suggest about the solution behavior?
   Answer: Ion pairing effects become significant, reducing the effective ionic strength and increasing the observed activity coefficient above predicted values When you encounter activity coefficient deviations from Debye-Hückel theory, you're dealing with limitations of electrolyte solution models at higher concentrations. The Debye-Hückel limiting law assumes complete ionic dissociation and treats ions as point charges, but real solutions behave differently as concentration increases. The key observation here is that the experimental activity coefficient (0.875) is higher than the predicted value (0.823). This tells you that the solution is behaving more ideally than the limiting law predicts, which seems counterintuitive but points to a specific phenomenon. Answer A correctly identifies ion pairing as the cause. At concentrations above ~0.01 M, oppositely charged ions begin forming ion pairs or contact ion pairs. This reduces the number of free ions in solution, effectively lowering the ionic strength below what you'd calculate from the nominal concentration. Since activity coefficients increase as ionic strength decreases (approaching 1.0 for ideal behavior), ion pairing leads to higher observed activity coefficients. Answer B is incorrect because excluded volume effects would make the solution behave less ideally, lowering activity coefficients below predicted values. Answer C focuses on experimental error rather than fundamental solution behavior—while temperature control matters, systematic deviations at specific concentrations indicate real physical phenomena. Answer D similarly attributes the discrepancy to instrumental issues rather than solution chemistry. Remember: when experimental activity coefficients are higher than Debye-Hückel predictions at moderate concentrations, think ion pairing first. This is a fundamental limitation that appears consistently in electrolyte solutions.
5. In a solution containing both NaCl and KCl with total ionic strength of 0.2 M, the activity coefficient of Na⁺ is found to be 0.70. Using the principle of ionic strength, what can be concluded about the activity coefficient of K⁺ in the same solution?
   Answer: It will be approximately 0.70 because ionic strength effects dominate over specific ion interactions According to Debye-Hückel theory, the activity coefficient depends primarily on ionic strength and charge, not on the specific identity of the ion. Since Na⁺ and K⁺ have the same charge (+1) and experience the same ionic strength environment, their activity coefficients should be very similar, approximately 0.70. Choice A is too absolute about exact equality. Choice C overemphasizes specific ion effects. Choice D is incorrect as ionic strength captures the essential information.
6. The activity coefficient of a weak electrolyte HA in solution depends on both its degree of dissociation α and the ionic strength created by its dissociation. For a 0.10 M solution of HA with α = 0.15 and considering that γ±(HA) = 0.92 for the ionic species, what is the activity of undissociated HA molecules if their activity coefficient is γ(HA) = 1.08?
   Answer: a(HA) = 0.092 because most of the acid remains undissociated with activity coefficient correction When dealing with weak electrolyte solutions, you need to account for both dissociation equilibrium and activity coefficients. The activity of undissociated molecules equals their molar concentration times their activity coefficient. For this weak acid HA with degree of dissociation α = 0.15, only 15% of the original 0.10 M dissociates into ions. This means 85% remains as undissociated HA molecules, giving a concentration of undissociated HA = 0.10 M × (1 - 0.15) = 0.085 M. The activity of undissociated HA is then: $a(\text{HA}) = [\text{HA}] × γ(\text{HA}) = 0.085 \text{ M} × 1.08 = 0.092$ Answer A correctly identifies this value and properly explains that most acid remains undissociated with the activity coefficient providing a small correction. Answer B (0.108) incorrectly uses the full initial concentration (0.10 M) without accounting for the 15% that dissociated. Answer C (0.085) gives only the concentration of undissociated molecules but forgets to apply the activity coefficient correction. Answer D (0.100) ignores both dissociation and the activity coefficient, treating the system as if it were completely ideal and undissociated. Study tip: For weak electrolyte problems, always work in steps: first calculate how much remains undissociated using (1 - α), then apply the activity coefficient. Don't confuse the activity coefficient for ions (γ±) with that for neutral molecules (γ).
7. A solution of 0.100 M NaCl is prepared in water at 25°C. The activity coefficient of NaCl at this concentration is 0.778. If this solution is then diluted to 0.0500 M while maintaining constant temperature, and the activity coefficient at the new concentration is 0.820, what is the ratio of the activity of NaCl in the diluted solution to its activity in the original solution?
   Answer: 0.528 This question tests your understanding of activity, which differs from concentration in real solutions due to ion-ion interactions. Activity (a) equals the product of molarity (M) and activity coefficient (γ): $a = γ × M$. To find the ratio of activities, you need to calculate the activity in each solution. In the original 0.100 M solution: $a_1 = 0.778 × 0.100 = 0.0778$. In the diluted 0.0500 M solution: $a_2 = 0.820 × 0.0500 = 0.0410$. The ratio of activities is: $\frac{a_2}{a_1} = \frac{0.0410}{0.0778} = 0.527$, which rounds to 0.528. Notice that the activity coefficient increased from 0.778 to 0.820 upon dilution. This happens because ionic interactions weaken as ions become more separated in dilute solutions, making the solution behave more ideally (γ approaches 1.0). Looking at the wrong answers: B) 0.410 represents just the activity of the diluted solution, not the ratio. C) 0.500 would be the ratio if activity coefficients were ignored and you simply used the concentration ratio (0.0500/0.100). D) 0.615 doesn't correspond to any meaningful calculation in this problem. Study tip: Always remember that activity = γ × M, and don't confuse the individual activity values with their ratio. Activity coefficients typically increase (approach 1.0) as ionic solutions become more dilute, which is why the ratio isn't simply the concentration ratio.

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