Problem 80 - Chemistry The Central Science

Question

For the equilibrium $$\mathrm{PH}_{3} \mathrm{BCl}_{3}(s) \rightleftharpoons \mathrm{PH}_{3}(g)+\mathrm{BCl}_{3}(g)$$ \(K_{p}=0.052\) at \(60^{\circ} \mathrm{C}\) (a) Calculate \(K_{C}\) (b) After 3.00 \(\mathrm{g}\) of solid \(\mathrm{PH}_{3} \mathrm{BCl}_{3}\) is added to a closed 1.500 -L. vessel at \(60^{\circ} \mathrm{C}\) , the vessel is charged with 0.0500 \(\mathrm{g}\) of \(\mathrm{BCl}_{3}(g) .\) What is the equilibrium concentration of \(\mathrm{PH}_{3} ?\)

Step-by-Step Solution

Answer

The equilibrium concentration of PH3 is approximately 0.00107 M.

Step 1: Convert Kp to Kc

To convert Kp to Kc, we'll use the relationship between Kp and Kc: \[K_{p} = K_{c}(RT)^{\Delta n}\] where R is the ideal gas constant (0.08206 L⋅atm/mol⋅K), T is the temperature in Kelvin, and Δn is the difference in moles of gaseous products and reactants. In this case: - Δn = (1 mole of PH3 + 1 mole of BCl3) - 0 moles of PH3BCl3 = 2 - 0 = 2 - T = 60°C + 273.15 = 333.15 K - Kp = 0.052 Substitute these values into the equation and solve for Kc: \[0.052 = K_{c}(0.08206 \times 333.15)^{2}\] \[K_{c} = \frac{0.052}{(0.08206 \times 333.15)^{2}}\] Now calculate the value of Kc:

Step 2: Calculate Kc

By calculating the expression from Step 1, we get: \[K_{c} \approx 8.71 \times 10^{-6}\]

Step 3: Set up an ICE table

To find the equilibrium concentration of PH3, we first set up an ICE (Initial, Change, Equilibrium) table: | | PH3(g) | BCl3(g) | PH3BCl3(s) | |-------|--------|---------|------------| | Initial | 0 | 0.0500/1.5 | 3/1.5 | | Change | +x | +x | -x | | Equilibrium | x | 0.0500/1.5 + x | 3/1.5 - x |

Step 4: Write the expression for Kc

Now, we'll write the Kc expression for the reaction: \[K_{c} = \frac{[\mathrm{PH}_{3}][\mathrm{BCl}_{3}]}{[\mathrm{PH}_{3}\mathrm{BCl}_{3}]}\] Substitute the equilibrium concentrations from the ICE table: \[8.71 \times 10^{-6} = \frac{x(x + 0.0500/1.5)}{3/1.5 - x}\] Solve for x, which is the equilibrium concentration of PH3.

Step 5: Calculate the equilibrium concentration of PH3

After solving the equation from Step 4, we find: \[x \approx 0.00107\, M\] Therefore, the equilibrium concentration of PH3 is approximately 0.00107 M.

Key Concepts

Equilibrium Constant (Kc and Kp)

Chemical equilibrium involves the balance between the forward and reverse reactions in a closed system. A key part of understanding chemical equilibrium is the equilibrium constant, which comes in two varieties for gas reactions: \(K_c\) and \(K_p\). The equilibrium constant \(K_c\) is calculated using concentrations of reactants and products, while \(K_p\) is derived from partial pressures. These constants allow chemists to understand how much of the reactants and products will be present at equilibrium in a given chemical reaction.

To convert \(K_p\) to \(K_c\), we use the formula: \[ K_{p} = K_{c}(RT)^{\Delta n} \] * \(R\) is the ideal gas constant, typically \(0.08206\) L·atm/mol·K. * \(T\) is the temperature in Kelvin. * \(\Delta n\) is the change in moles of gas during the reaction: \(\text{moles of products} - \text{moles of reactants}\).
For the given exercise, the conversion from \(K_p\) to \(K_c\) involves a temperature of 333.15 K and a \(\Delta n\) of 2. Learning to transform \(K_p\) to \(K_c\) is crucial, as it allows the application of equilibrium principles to reactions taking place in gaseous environments.

ICE Table Method

When addressing chemical equilibrium problems, the ICE table method is a systematic approach to figure out the concentrations of species at equilibrium. ICE stands for Initial, Change, and Equilibrium, representing the different stages in a chemical reaction. Here's how you break it down:

* **Initial**: Start with the initial concentrations or pressures of reactants and products before the reaction begins. * **Change**: Determine how the concentrations change as the system reaches equilibrium. Typically, changes are indicated by \(+x\) or \(-x\). * **Equilibrium**: Sum the initial amounts and changes to find the final equilibrium concentrations or pressures.

In the problem above, an ICE table is used to track the changes in concentrations of \(\text{PH}_3\) and \(\text{BCl}_3\). The starting point (itial amounts) involves specific amounts of solids and gases. As the reaction progresses to equilibrium, the changes \(+x\) or \(-x\) help find the final values. By solving the resulting equations, you can determine unknown concentrations like the equilibrium concentration of \(\text{PH}_3\), giving valuable insights into the reaction's progress.

Gas Laws in Chemistry

Gas laws are essential to understanding chemical reactions involving gases. They relate the quantities of gas, pressure, volume, temperature, and the number of particles, providing vital context when evaluating reactions. Here's a quick overview of the useful gas laws:

* **Ideal Gas Law**: \(PV = nRT\), which relates pressure \(P\), volume \(V\), and temperature \(T\) of an ideal gas with the number of moles \(n\) and the ideal gas constant \(R\). * **Boyle’s Law**: Concerns pressure and volume \((P_1V_1 = P_2V_2)\) at a constant temperature. * **Charles’s Law**: Focuses on volume and temperature \((V_1/T_1 = V_2/T_2)\) at constant pressure. * **Avogadro’s Law**: Links volume to the number of moles \((V_1/n_1 = V_2/n_2)\) at constant temperature and pressure.

Understanding gas laws aids students in comprehending how changes in conditions affect gaseous reactions. They come into play significantly when working with equilibrium constants in gas reactions, as they allow conversions and calculations such as transforming \(K_p\) to \(K_c\), considering moles of gases involved. As reactions under equilibrium typically involve changes in pressure and temperature, grasping these laws helps predict how a system may shift upon external changes.