Question

Starting with \(\mathrm{SO}_{3}(\mathrm{g})\) at \(1.00 \mathrm{atm},\) what will be the total pressure when equilibrium is reached in the following reaction at \(700 \mathrm{K} ?\) \(2 \mathrm{SO}_{3}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \quad K_{\mathrm{p}}=1.6 \times 10^{-5}\)

Short answer

The total equilibrium pressure of the system at \(700 \mathrm{K}\) is 0.9984 atm.

Step-by-step solution

Define Initial Conditions and Reactions

Initially, there is 1 atm of \(SO_{3}\) and no \(SO_{2}\) or \(O_{2}\). When the reaction reaches equilibrium, some \(SO_{3}\) will have converted into \(SO_{2}\) and \(O_{2}\). Define \(x\) to be the change in pressure of \(SO_{3}\), so the change in \(SO_{3}\) is -2x (since it is being consumed), and the changes in \(SO_{2}\) and \(O_{2}\) are +2x and +x, respectively (since they are being produced).

Formulate the ICE Table

Write down the initial, change, and equilibrium expressions for all species involved: initial pressures are \(SO_{3}: 1, SO_{2}: 0, O_{2}: 0\), changes in pressures are \(SO_{3}: -2x, SO_{2}: +2x, O_{2}: +x\), and equilibrium pressures are \(SO_{3}: 1-2x, SO_{2}: 2x, O_{2}: x\).

Formulate the Equilibrium Constant Expression

Next, write down the equilibrium constant \(K_{\mathrm{p}}\) expression based on the balanced chemical equation: \(K_{\mathrm{p}} = \frac{{[SO_{2}]^2[O_{2}]}}{{[SO_{3}]^2}}\) and substitute the equilibrium pressures from the ICE table, yielding \(1.6 \times 10^{-5} = \frac{{(2x)^2 \cdot x}}{{(1-2x)^2}}\).

Solve for x

The equation in Step 3 yields a cubic equation, which can be challenging to solve algebraically. However, making an assumption that x << 1 (since \(K_{\mathrm{p}}\) is very small indicating that not much reactant is converted to product), the equation simplifies to: \(1.6 \times 10^{-5} = 4x^3\), which can be solved to find x = 0.0012.

Calculate Final Pressure at Equilibrium

Substitute x = 0.0012 into the expressions for total pressure: \(P_{total} = P_{SO_{3}} + P_{SO_{2}} + P_{O_{2}} = (1 - 2 \cdot 0.0012) + 2 \cdot 0.0012 + 0.0012 = 0.9984 \: atm\).

Key concepts

Equilibrium Constant (Kp)

The equilibrium constant (K_p) is a number that reflects the ratio of the concentrations of products to reactants at equilibrium in a chemical reaction that involves gases. In the given exercise, the equilibrium constant helps determine the extent to which the reaction:

• 2 SO_3(g) \(\rightleftharpoons\) 2 SO_2(g) + O_2(g)

at 700 K will proceed.The equilibrium constant K_p is specific for each reaction at a given temperature.It is expressed as the ratio of partial pressures of the gaseous products raised to their stoichiometric coefficients divided by the partial pressures of the reactants raised to theirs:\[K_p = \frac{{[SO_2]^2[O_2]}}{{[SO_3]^2}}\]Where:

• [SO_2], [O_2], and [SO_3] are the equilibrium partial pressures of the gases.
• Each term is raised to the power of its stoichiometric coefficient above its respective gas.

In this exercise, with a given K_p of 1.6 \times 10^{-5}, the small value indicates that at equilibrium, the reactants (SO_3) are predominant. Understanding K_p allows you to predict how changes in conditions will affect the equilibrium state.

ICE Table

An ICE Table is a useful tool for visualizing the changes in concentration or pressure in a chemical reaction as it approaches equilibrium. The 'ICE' in ICE Table stands for Initial, Change, and Equilibrium:

• Initial: The initial conditions before the reaction starts, such as pressure or concentration.
• Change: The changes that occur as equilibrium is established. These changes are often expressed in terms of a variable like \(x\).
• Equilibrium: The final conditions once the reaction has reached equilibrium.

In the given reaction of SO_3 converting to SO_2 and O_2:

• Initially, there is 1 atm of SO_3, and no SO_2 or O_2.
• At equilibrium, the pressure of SO_3 changes by -2x, since each molecule of SO_3 produces two molecules of SO_2.The changes for SO_2 and O_2 are +2x and +x, respectively.

Using an ICE table simplifies the process of writing out the equilibrium expressions and helps in calculating pressures or concentrations needed to determine the equilibrium state.

Le Chatelier's Principle

Le Chatelier's Principle is a key concept in understanding how changing conditions affect the equilibrium of a chemical reaction. It states that if a system at equilibrium is subjected to a change in temperature, pressure, or concentration, the system will adjust in such a way as to counteract that change and restore a new equilibrium. For the given reaction:

• If the pressure is increased by adding more SO_3 , the system will respond by converting some of the extra SO_3 into SO_2 and O_2 , shifting equilibrium to the right.
• If the temperature changes, it's essential to consider the endothermic or exothermic nature of the reaction to predict the direction of the shift.

Given a small K_p in this situation, it suggests the reaction is reactant-favored, meaning more SO_3 will be present than products at equilibrium. Therefore, according to Le Chatelier, changes that favor the formation of products will shift the equilibrium position noticeably.

Gas Phase Reactions

Gas phase reactions involve reactants and products that exist in the gaseous state. This means the reaction's behavior is influenced by changes in volume, pressure, and temperature. Each of these variables can impact the equilibrium.

• In reactions involving gases, the pressure can play a crucial role because all reactants and products are subject to compressibility and expansion with temperature changes.
• Partial pressures are used instead of concentrations, which leads to the development of the equilibrium constant K_p, which specifically relates to partial pressures.
• Some reactions expand in volume (produce more gas) while others compress (produce fewer gas moles), and changes in pressure can push the equilibrium position according to proposed shifts.

For this specific reaction, we see gaseous SO_3 decomposing into other gases, and because all components are in the gas phase, the use of the ideal gas law and conversion between volume, moles, and pressure are integral parts of understanding how the reaction behaves and reaches equilibrium.