Question

The equilibrium constant \(K_{\mathrm{p}}\) for the reaction $$ \mathrm{CCl}_{4}(g) \rightleftharpoons \mathrm{C}(s)+2 \mathrm{Cl}_{2}(g) $$ at \(700^{\circ} \mathrm{C}\) is \(0.76 .\) Detemine the initial pressure of carbon tetrachloride that will produce a total equilibrium pressure of \(1.20 \mathrm{~atm}\) at \(700^{\circ} \mathrm{C}\).

Short answer

The initial pressure of carbon tetrachloride that will produce a total equilibrium pressure of 1.20 atm at 700°C is 1.80 atm.

Step-by-step solution

Set up an ICE table

An ICE table allows us to keep track of the Initial, Change, and Equilibrium concentrations or pressures of the species in a chemical reaction. For this reaction, we have: \(CCl_4 (g) \rightleftharpoons C (s) + 2Cl_2 (g)\) Let the initial pressure of the CCl4 be P. Since we are solving for the initial pressure, we'll leave it as P, and the initial pressure of Cl2 will be 0. The change in pressure for C's solid does not influence the equilibrium constant, so we do not need to track it. ICE table: CCl4 C Cl2 Initial P 0 0 Change -x - 2x Equilibrium P-x - 2x At equilibrium, the total pressure is given as 1.20 atm.

Write the equilibrium constant expression

The equilibrium constant expression can be written using the equilibrium pressures. Since the carbon is in solid form, it is not included in the equilibrium constant expression. Kp is given as 0.76. Kp = (\( \frac {[Cl_2]^2} {[CCl_4]} \)) = 0.76

Substitute the equilibrium pressures into the equilibrium constant expression and solve for x

Now we will substitute the equilibrium pressures from the ICE table into the equilibrium constant expression. 0.76 = (\( \frac {(2x)^2} {P-x} \))

Solve for the initial pressure, P

Given that the total equilibrium pressure is 1.20 atm, we can write an equation for it. 1.20 atm = (P-x) + 2x Now, we can solve for x from the equilibrium constant expression, and substitute it into the equation for the total equilibrium pressure to get the initial pressure, P. From the equilibrium constant expression: x = 0.60 P = 1.80 atm The initial pressure of carbon tetrachloride that will produce a total equilibrium pressure of 1.20 atm at 700°C is 1.80 atm.

Key concepts

ICE table

An ICE table is a valuable tool for managing the initial, change, and equilibrium states of substances in a chemical reaction. It helps organize information systematically for better analysis.
In the given problem, we are dealing with the decomposition of carbon tetrachloride (\(CCl_4\)) into chlorine gas (\(Cl_2\)) and carbon (\(C\)) as a solid at equilibrium. We use the ICE table to record the changes each substance undergoes as the system reaches stability:

• **I**: Initial pressures of each reactant and product.
• **C**: Changes in pressure as the system shifts towards equilibrium.
• **E**: Equilibrium pressures after the reaction has progressed.

This method makes it easy to visualize the effects of pressure changes and solve for unknown variables like the initial pressure of carbon tetrachloride.

equilibrium pressure

In the context of chemical equilibrium, equilibrium pressure refers to the pressure exerted by the gases when the system is at equilibrium. This is when the rates of the forward and reverse reactions are equal, making the net change zero.
For the decomposition of carbon tetrachloride reaction, understanding equilibrium pressure is crucial. At equilibrium, the total pressure of the gases involved was given as 1.20 atm.
This total equilibrium pressure is the sum of the partial pressures: the pressure of the remaining \(CCl_4\) and the pressure of the \(Cl_2\) produced. This value provides a means to solve for unknown pressures using the equilibrium constant (\(K_p\)).

chemical equilibrium

Chemical equilibrium occurs when a chemical reaction proceeds such that the concentrations of reactants and products remain constant over time. This state signifies a balance in the chemical process, where the forward and reverse reactions happen at the same rate.
In the case of carbon tetrachloride decomposition, we are given that the reaction has reached equilibrium at a specific temperature of 700°C. At this stage, using the equilibrium constant (\(K_p = 0.76\)), we can calculate how much \(CCl_4\) is needed initially to achieve a specific pressure of 1.20 atm when the system is at balance.
The stability attained here is central to solving equilibrium problems and predicting the outcomes of temperature or pressure changes.

carbon tetrachloride

Carbon tetrachloride (\(CCl_4\)) is a compound that readily participates in decomposition reactions. In this context, it breaks down into solid carbon and chlorine gas at high temperatures, such as 700°C.
This compound, being volatile, directly influences the equilibrium dynamics of such chemical processes. Its initial pressure is a critical variable in solving equilibrium problems like the one posed here.
By using the equation that defines its decomposition, along with the given equilibrium constant, you can determine the necessary starting pressure to reach the desired total equilibrium pressure.