Question

Phosphorous pentachloride dissociates as follows, in a closed reaction vessel. \(\mathrm{PCI}_{5}(\mathrm{~g}) \longrightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\). If total pressure at equilibrium of the reaction mixture is \(\mathrm{P}\) and degree of dissociation of \(\mathrm{PC} 1_{5}\) is \(x\), the partial pressure of \(\mathrm{PCl}_{3}\) will be: (a) \(\left(\frac{x}{(x+1)}\right) \mathrm{P}\) (b) \(\left(\frac{2 x}{(x-1)}\right) \mathrm{P}\) (c) \(\left(\frac{x}{(x-1)}\right) \mathrm{P}\) (d) \(\left(\frac{x}{(1-x)}\right) \mathrm{P}\)

Short answer

(a) \(\left(\frac{x}{1+x}\right) P\)

Step-by-step solution

Understanding the Problem

We need to determine the partial pressure of \(\mathrm{PCl}_3\) at equilibrium in terms of total pressure \(P\) and degree of dissociation \(x\). Given the reaction \(\mathrm{PCl}_5(g) \rightarrow \mathrm{PCl}_3(g) + \mathrm{Cl}_2(g)\), let's work through the changes in moles at equilibrium.

Initial Moles and Changes

Initially, assume 1 mole of \(\mathrm{PCl}_5\) is present, which starts to dissociate. Let \(x\) be the fraction that dissociates, so at equilibrium, \((1 - x)\) moles of \(\mathrm{PCl}_5\) remain, while \(x\) moles of \(\mathrm{PCl}_3\) and \(x\) moles of \(\mathrm{Cl}_2\) are formed.

Total Moles at Equilibrium

The total moles of gas at equilibrium are \((1 - x) + x + x = 1 + x.\) These result from the undissociated \(\mathrm{PCl}_5\) and the newly formed \(\mathrm{PCl}_3\) and \(\mathrm{Cl}_2\).

Relating Partial Pressure to Total Pressure

The total pressure \(P\) of the mixture is related to the total moles of gas, \(1 + x\). Since the pressure is proportional to the amount of gas (ideal gas law assumption), the partial pressure of \(\mathrm{PCl}_3\) is equal to the fraction of \(\mathrm{PCl}_3\) moles over total moles, times \(P\).

Calculating Partial Pressure of \(\mathrm{PCl}_3\)

At equilibrium, \(x\) moles of \(\mathrm{PCl}_3\) are present out of a total of \(1 + x\) moles. Thus, the partial pressure of \(\mathrm{PCl}_3\) is \(\frac{x}{{1 + x}} \times P\).

Choose the Correct Option

The expression for the partial pressure of \(\mathrm{PCl}_3\) matches with option (a): \(\left(\frac{x}{1+x}\right) P\).

Key concepts

Partial Pressure

To fully understand what partial pressure means, it's important to think about a gas mixture. In a container with multiple gases, each type of gas contributes to the total pressure. This contribution is the partial pressure of that gas.

Partial pressure is the pressure a gas would exert if it were the only gas present in the container. In equations, we often denote partial pressure with a lowercase letter representing the gas, like \(p_\text{gas}\). It's a helpful way to understand how each gas behaves in a mix.

• **Total Pressure**: The sum of partial pressures of all gases in the mix.
• **Partial Pressure Formula**: For a particular gas, it's calculated with the equation \(p_\text{gas} = \text{mole fraction of gas} \times \text{total pressure}\).
• **Mole Fraction**: This is the ratio of the moles of one gas to the total moles of gas in the mixture.

Understanding partial pressure is critical when dealing with reactions involving gases, as it helps predict how gases will react with one another or shift the position of equilibrium in a reaction.

Degree of Dissociation

Degree of dissociation tells us how much a compound breaks down into its components in a chemical reaction. Specifically, it refers to the fraction of original molecules that split into products.

Imagine starting with 1 mole of a compound that can decompose or split apart. If 0.3 moles of it dissociate, the degree of dissociation is 0.3. This value is often expressed as a percentage by multiplying by 100. This concept is crucial in equilibrium reactions where knowing how much of a substance has reacted can determine the concentration of products.

• **Symbol**: Usually denoted by \(x\) or \(\alpha\).
• **Equation**: \(\text{Degree of Dissociation} = \frac{\text{moles dissociated}}{\text{initial moles}}\).
• **Impact on Reaction**: Determines extent to which reactants are converted into products at equilibrium.

Dissociation significantly affects calculations involving equilibrium constants and partial pressures, as seen in reactions like that of phosphorus pentachloride.

Ideal Gas Law

The Ideal Gas Law is a fundamental relationship between the four variables of a gas: pressure, volume, temperature, and amount in moles. The formula is expressed as \(PV = nRT\), linking these variables together.

Here, \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. This law assumes gases behave ideally, meaning they occupy no space and exert no forces on each other. While real gases can deviate from this behavior, the ideal gas law provides a close approximation under standard temperature and pressure.

• **Gas Constant (R)**: Typically valued at 0.0821 L atm/mol K in these units.
• **Applications**: Used to predict how a gas will react to changes in environment or conditions.
• **Limitations**: Assumption of no intermolecular interactions isn't perfect for real gases.

The Ideal Gas Law often assists in solving for one unknown variable when the other three are known, especially useful in equilibrium problems.

Phosphorus Pentachloride

Phosphorus pentachloride, \(\text{PCl}_5\), is a compound of phosphorus and chlorine and is a common inorganic compound with various industrial applications. It exists as a white crystalline solid and is known to decompose into \(\text{PCl}_3\) and \(\text{Cl}_2\) upon heating.

This behavior is useful in demonstrating concepts of chemical equilibrium as it readily dissociates into its components. Understanding phosphorus pentachloride is important for exercises involving degree of dissociation, equilibrium constants, and pressure calculations.

• **Reactivity**: Used for chlorination processes, it acts as a chlorinating agent.
• **Physical Properties**: Solid at room temperature with a distinct white crystalline appearance.
• **Decomposition**: It decomposes especially at higher temperatures, making it a good example for learning equilibrium dynamics.

Phosphorus pentachloride's role in equilibrium reactions showcases how temperature and pressure changes can shift equilibria, making it crucial in industrial chemistry.

Equilibrium Constant

Chemical reactions reach a state where the concentrations of reactants and products become constant. This is known as equilibrium, and the equilibrium constant \(K\) quantifies the ratio of product concentrations to reactant concentrations at this point.

This constant is unique for every reaction and is determined by the specific balance of products and reactants when no further change occurs in the system. Mathematically, it's expressed in the form of \(K_c\) for concentrations and \(K_p\) for partial pressures.

• **Formula**: For a general reaction \(aA + bB \leftrightarrow cC + dD\), \(K_c\) is \(\frac{[C]^c[D]^d}{[A]^a[B]^b}\).
• **Influence**: At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.
• **Importance**: Helps predict the direction of the reaction, extent of reaction and effects of changing conditions.

By understanding equilibrium constants, students can predict how varying conditions like temperature or pressure will affect a chemical reaction's position.