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Chapter 7: Problem 80

The ratio of \(\mathrm{Kp} / \mathrm{Kc}\) for the reaction \(\mathrm{SO}_{2}(\mathrm{~g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{SO}_{3}(\mathrm{~g})\) is (a) \((\mathrm{RT})^{-1 / 2}\) (b) \((\mathrm{RT})^{1 / 2}\) (c) \(\mathrm{RT}\) (d) 1

Short Answer

Expert verified
The ratio \( \frac{K_p}{K_c} \) is \( (RT)^{-1/2} \), corresponding to option (a).

Step by step solution

01

Identify Variables in Ideal Gas Constant Equation

The ratio \( \frac{K_p}{K_c} \) for a gaseous reaction is given by the formula \( K_p = K_c (RT)^{\Delta n} \), where \( \Delta n \) is the change in the number of moles of gas (products - reactants). Here, \( R \) is the ideal gas constant and \( T \) is the temperature in Kelvin.
02

Compute Delta n (Change in Moles)

For the reaction \( \mathrm{SO_2(g)} + \frac{1}{2}\mathrm{O_2(g)} \rightleftharpoons \mathrm{SO_3(g)} \), calculate \( \Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} = 1 - \left( 1 + \frac{1}{2} \right) = 1 - 1.5 = -0.5 \).
03

Substitute Delta n into the Ratio Formula

Using \( \Delta n = -0.5 \), substitute this into the equation \( \frac{K_p}{K_c} = (RT)^{\Delta n} \) to get \( \frac{K_p}{K_c} = (RT)^{-0.5} \).
04

Determine Correct Answer Choice

Compare the derived expression \( (RT)^{-0.5} \) with the provided options. The option that matches is (a) \( (RT)^{-1/2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilibrium Constants
In chemical reactions, especially those involving gases, understanding equilibrium constants is crucial. These constants help predict how a reaction will proceed under certain conditions. There are two main types:
  • \(K_c\): The equilibrium constant for gases expressed in concentrations. It is calculated using molar concentrations of the reactants and products at equilibrium.
  • \(K_p\): The equilibrium constant for gases expressed in partial pressures. It is used when dealing with reactions carried out in the gas phase.
The relationship between \(K_c\) and \(K_p\) is dictated by the formula \(K_p = K_c(RT)^{\Delta n}\), where \(R\) is the ideal gas constant, \(T\) is temperature, and \(\Delta n\) is the change in moles of gas. Understanding this relationship helps in predicting how changes in pressure and temperature can affect a gaseous reaction's equilibrium.
Gas Laws
The behavior of gases under different conditions is described by gas laws. These laws are fundamental for understanding reactions involving gases.
  • Boyle鈥檚 Law: This law states that pressure and volume are inversely proportional, assuming temperature and the number of gas moles remain constant. It can be expressed as \(P_1V_1 = P_2V_2\).
  • Charles's Law: Volume and temperature are directly proportional, assuming pressure and the number of moles remain constant. Expressed as \(V_1/T_1 = V_2/T_2\).
  • Avogadro鈥檚 Law: Volume and the number of moles of the gas are directly proportional at a constant temperature and pressure. It is given by \(V \propto n\).
  • Ideal Gas Law: Combines all the simple gas laws into one equation: \(PV = nRT\).
Understanding these laws and the Ideal Gas Law is important because they are the foundation of many calculations involving gases in chemistry.
Reaction Quotients
The reaction quotient, \(Q\), is a pivotal concept when analyzing reactions. It's a measure for the relative quantities of products and reactants during a reaction at any given point in time:
  • If \(Q = K\), the system is at equilibrium.
  • If \(Q < K\), the forward reaction is favored, meaning more products will form.
  • If \(Q > K\), the reverse reaction is favored, meaning the reaction will proceed to form more reactants.
Calculating \(Q\) helps in determining the shift in equilibrium position, especially vital when the system undergoes changes in conditions such as pressure or concentration.
Ideal Gas Constant
The ideal gas constant, often symbolized as \(R\), is a crucial parameter in the Ideal Gas Law. It serves to bridge various units used in gas calculations, and its value depends on the units of pressure, volume, and temperature:
  • The most common value of \(R\) is 8.314 J/(mol路K), used when pressure is in pascals and volume in cubic meters.
  • In contexts where pressure is in atmospheres, its value is taken as 0.0821 L路atm/(mol路K).
The universal nature of \(R\) makes it suitable across multiple disciplines, ensuring consistency in calculations involving gas behaviors. Its role in the formulation \(K_p = K_c(RT)^{\Delta n}\) highlights its importance in relating equilibrium constants for gaseous reactions.

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Most popular questions from this chapter

The ratio of \(K_{p} / K_{c}\) for the reaction \(\mathrm{CO}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{CO}_{2}(\mathrm{~g})\) is (a) \((\mathrm{RT})^{1 / 2}\) (b) \((\mathrm{RT})^{-1 / 2}\) (c) RT (d) 1

The relation between \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) for the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g})\) is (a) \(\mathrm{K}_{\mathrm{n}}=\mathrm{K}_{\mathrm{c}}(\mathrm{RT})^{-1}\) (b) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}}\) (c) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}} /(\mathrm{RT})^{2}\) (d) \(K_{p}^{p}=K_{c} / R T\)

A saturated solution of non-radioactive sugar was taken and a little radioactive sugar was added to it. A small amount of it gets dissolved in solution and an equal amount of sugar was precipitated. This proves (a) the equilibrium has been established in the solution (b) radioactive sugar can displace non-radioactive sugar from its solution. (c) Equilibrium is dynamic in nature (d) none of the above

For the reaction, \(\mathrm{H}_{2}+\mathrm{I}_{2} \rightleftharpoons 2 \mathrm{HI}\), the equilibrium concentrations of \(\mathrm{H}_{2}, 1_{2}\) and \(\mathrm{HI}\) are 8,3 and \(28 \mathrm{~mol} \mathrm{~L}^{-1}\) respectively. Equilibrium constant of the reaction is (a) \(32.67\) (b) \(31.67\) (c) \(34.67\) (d) \(36.67\)

The equilibrium constant of a reaction is 300 . If the volume of reaction flask is tripled, the equilibrium constant is (a) 300 (b) 600 (c) 900 (d) 100
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