Hey guys! Today, we're diving into a fascinating chemistry problem involving the equilibrium constant and Gibbs free energy. We'll be looking at the reaction between silver ions and ammonia, and figuring out the standard free energy change. This is a classic example that pops up in many chemistry courses, and understanding it can really boost your grasp of thermodynamics and equilibrium. So, let's break it down step by step!
Understanding the Reaction: Ag+(aq) + 2NH3(aq) ⇌ Ag(NH3)2+(aq)
The reaction we're focusing on is the formation of the diamminesilver(I) complex ion. Silver ions ($Ag^+$, which are positively charged silver atoms floating around in water) love to hang out with ammonia ($NH_3$, a nitrogen atom with three hydrogen atoms attached). When silver ions meet ammonia in a watery environment (that's what the (aq) means, by the way, it stands for aqueous), they team up to form a complex ion called diamminesilver(I) ($Ag(NH_3)_2^+$, which is a silver ion with two ammonia molecules attached to it). This is a reversible reaction, meaning it can go both ways: the silver ion and ammonia can combine, and the diamminesilver(I) complex can also break apart back into silver ions and ammonia. The double arrow (⇌) in the reaction equation tells us it's a reversible process.
Now, here's where the equilibrium constant, K, comes into play. It's a number that tells us how much the reaction favors the products (the diamminesilver(I) complex) over the reactants (silver ions and ammonia) at equilibrium. Equilibrium is a state where the rate of the forward reaction (forming the complex) equals the rate of the reverse reaction (breaking the complex). In our case, the equilibrium constant, K, is given as $1.7 imes 10^7$ at $25^{\circ} C$. This is a huge number, which means the reaction strongly favors the formation of the diamminesilver(I) complex. In simpler terms, at equilibrium, there will be way more diamminesilver(I) complex hanging around than silver ions and ammonia.
To really understand what's going on, let's think about it like a dance party. Imagine silver ions are shy dancers, and ammonia molecules are the friendly folks who can get them on the dance floor. When two ammonia molecules latch onto a silver ion, they form a dancing pair – the diamminesilver(I) complex. A large equilibrium constant is like saying this party has a super popular dance, so most of the shy dancers end up getting paired up and dancing. This complex formation is a crucial concept in coordination chemistry, which deals with how metal ions interact with ligands (like ammonia in this case) to form complex ions. The stability of these complexes is vital in various applications, from chemical analysis to biological systems. For example, diamminesilver(I) ions are used in certain silver plating processes and have antimicrobial properties. Understanding the equilibrium constant helps us predict and control the extent of complex formation, which is essential for these applications. So, grasping this concept is not just about solving this problem; it's about unlocking a deeper understanding of how chemical reactions work in real-world scenarios. And that's pretty cool, right?
Connecting Equilibrium Constant (K) and Gibbs Free Energy (ΔG°)
Okay, so we know the equilibrium constant (K), but what about the Gibbs free energy change (ΔG°)? Gibbs free energy is like the ultimate indicator of spontaneity in a chemical reaction. It tells us whether a reaction will happen on its own (spontaneous) or if we need to put in some extra energy to make it go (non-spontaneous). The Gibbs free energy change (ΔG°) specifically refers to the change in free energy under standard conditions, which usually means 25°C (298 K) and 1 atmosphere of pressure. A negative ΔG° means the reaction is spontaneous – it's like a ball rolling downhill, it happens naturally. A positive ΔG° means the reaction is non-spontaneous – it's like pushing a ball uphill, you need to put in effort. And a ΔG° of zero means the reaction is at equilibrium – it's like the ball is on a flat surface, it's not going anywhere unless something pushes it.
Now, here's the really cool part: the equilibrium constant (K) and the Gibbs free energy change (ΔG°) are directly related! There's a simple equation that connects them, and it's a cornerstone of chemical thermodynamics. This equation is:
Where:
- ΔG° is the standard Gibbs free energy change
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
- ln K is the natural logarithm of the equilibrium constant
This equation is super important because it bridges the gap between the equilibrium position of a reaction (how much product versus reactant we have at equilibrium) and the spontaneity of the reaction (whether it will happen on its own). It tells us that a large equilibrium constant (K) – meaning the reaction strongly favors product formation – corresponds to a large negative ΔG°, indicating a spontaneous reaction. Conversely, a small K (reaction favors reactants) means a positive ΔG°, indicating a non-spontaneous reaction. It's like the universe is constantly trying to minimize its free energy, so reactions that release energy (negative ΔG°) tend to happen on their own, while reactions that require energy (positive ΔG°) need an extra push.
Let’s think of it in another way, like a seesaw. On one side, we have the equilibrium constant (K), which tells us the relative amounts of reactants and products at equilibrium. On the other side, we have the Gibbs free energy change (ΔG°), which tells us whether the reaction is spontaneous. The equation ΔG° = -RT ln K is the fulcrum that connects these two sides. If K is large (the product side is heavier), ΔG° will be negative (the spontaneity side goes down). If K is small (the reactant side is heavier), ΔG° will be positive (the spontaneity side goes up). This connection is incredibly powerful because it allows us to predict the spontaneity of a reaction simply by knowing its equilibrium constant, or vice versa. And that's a pretty neat trick to have up your sleeve in the chemistry world!
Calculating ΔG° for the Reaction
Alright, now for the fun part: using the equation to calculate ΔG° for our reaction! We've got all the pieces of the puzzle. We know the equilibrium constant (K = $1.7 imes 10^7$), we know the temperature (T = $25^{\circ} C$), and we know the ideal gas constant (R = 8.314 J/(mol·K)). The only thing left to do is plug the values into the equation and crunch the numbers.
First, let's convert the temperature from Celsius to Kelvin because the equation requires temperature in Kelvin. To do this, we simply add 273.15 to the Celsius temperature:
T (K) = T (°C) + 273.15 T (K) = 25 + 273.15 T (K) = 298.15 K
Now we have all our units aligned, and we are ready to substitute the values into the equation:
Let's tackle the natural logarithm part first. The natural logarithm of $1.7 imes 10^7$ is approximately 16.64.
Now, we just multiply all those numbers together:
Since the answer options are given in kJ, let's convert joules to kilojoules by dividing by 1000:
So, the standard Gibbs free energy change (ΔG°) for this reaction at 25°C is approximately -41.383 kJ/mol. This negative value confirms that the reaction is indeed spontaneous under standard conditions, which makes sense given the large equilibrium constant we started with.
But wait, let's pause here for a second and think about what this number actually means in the context of our reaction. A ΔG° of -41.383 kJ/mol tells us that for every mole of diamminesilver(I) complex formed under standard conditions, 41.383 kilojoules of energy are released. This is a significant amount of energy, indicating a strong driving force for the reaction to proceed. It's like the reaction is energetically eager to happen! This high negative value is consistent with the large equilibrium constant, which tells us that the products (diamminesilver(I) complex) are much more stable than the reactants (silver ions and ammonia) at equilibrium. So, not only have we calculated ΔG°, but we've also gained a deeper understanding of the energetics and spontaneity of this fascinating reaction. And that's what chemistry is all about, right?
The Answer
Looking at the options provided, the closest value to our calculated ΔG° is:
A. -41.5 kJ
So, the correct answer is A. -41.5 kJ.
And there you have it, guys! We've successfully calculated the standard Gibbs free energy change (ΔG°) for the reaction between silver ions and ammonia, using the equilibrium constant (K) and the equation ΔG° = -RT ln K. We've not only solved the problem but also delved into the concepts of equilibrium, Gibbs free energy, and spontaneity, and how they're all interconnected. Remember, a large equilibrium constant means the reaction favors product formation, and a negative ΔG° means the reaction is spontaneous. These are key ideas in chemistry, and mastering them will take you far!
I hope this breakdown was helpful and made the concepts a little clearer. Chemistry can seem daunting at times, but breaking it down step by step, like we did here, can make it much more manageable. So, keep practicing, keep exploring, and most importantly, keep asking questions! You've got this!
Keywords
Equilibrium constant, Gibbs free energy, spontaneity, chemical reaction, silver ions, ammonia, diamminesilver(I) complex, thermodynamics, standard conditions.
