Chapter 16: Problem 29
Hydrogen \(\left(\mathrm{H}_{2}\right)\) is heated to \(3800 \mathrm{K}\) at a constant pressure of 5 atm. Determine the percentage of \(\mathrm{H}_{2}\) that will dissociate into H during this process.
Short Answer
Expert verified
Question: Calculate the percentage of hydrogen gas (H2) that dissociates into H atoms at a temperature of 3800 K and pressure of 5 atm.
Step by step solution
01
Calculate the equilibrium constant (K)
First, we need to find the equilibrium constant (K) for the dissociation of hydrogen gas at 3800K. The Van't Hoff equation can help us determine the proportional relationship between K and temperature (T):
$$\ln K = \frac{-\Delta H^\circ}{R} \left(\frac{1}{T} - \frac{1}{T^\circ}\right)$$
Where \(\Delta H^\circ\) is the standard enthalpy of the reaction, \(T\) is the given temperature, \(T^\circ\) is a reference temperature (298K for standard conditions), and \(R\) is the universal gas constant (\(8.314 \; \text{J/(mol K)}\)). The standard enthalpy of the dissociation of hydrogen gas is reported to be approximately \(\Delta H^\circ = 435.7 \, \text{kJ/mol}\). By plugging in these values, we can determine the equilibrium constant at 3800K.
02
Write the dissociation equation in terms of partial pressures and K
We can express the equilibrium condition for the dissociation of \(\mathrm{H}_2\) in terms of the partial pressures of the reactants and products:
$$K = \frac{P_\mathrm{H}^2}{P_{\mathrm{H}_2}}$$
Where \(P_\mathrm{H}\) is the partial pressure of H atoms, \(P_{\mathrm{H}_2}\) is the partial pressure of hydrogen gas, and K is the equilibrium constant calculated in Step 1.
03
Express partial pressures in terms of degree of dissociation (α)
Introduce the degree of dissociation (\(\alpha\)), which represents the fraction of hydrogen molecules that dissociate into H atoms. According to the reaction stoichiometry, the formation of 2 moles of H atoms requires 1 mole of \(\mathrm{H}_2\). Therefore, we can express the partial pressures as follows:
$$P_\mathrm{H} = 2(1 - \alpha) P_0\alpha \quad \text{and} \quad P_{\mathrm{H}_2} = P_0 - P_\mathrm{H}$$
Where \(P_0\) is the initial pressure of hydrogen gas, which is given as 5 atm.
04
Substitute the expressions for partial pressures into the equilibrium expression and solve for α
Replace the partial pressures in the equilibrium expression from Step 2 with the expressions from Step 3:
$$K = \frac{(2(1 - \alpha) P_0\alpha)^2}{P_0 - 2(1 - \alpha) P_0\alpha}$$
Now substitute the previously calculated equilibrium constant and initial pressure, then solve for the degree of dissociation (\(\alpha\)).
05
Calculate the percentage of H2 dissociated
To find the percentage of hydrogen gas that dissociates into H atoms, multiply the degree of dissociation (\(\alpha\)) by 100:
$$\text{Percentage of dissociation} = \alpha \times 100$$
Finally, report the percentage of \(\mathrm{H}_{2}\) that dissociates into H atoms at 3800K and 5 atm.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equilibrium Constant
The equilibrium constant, denoted as K, is crucial in understanding chemical reactions, particularly those in a state of dynamic equilibrium. It provides a numerical measure of a reaction's tendency to proceed towards products or reactants under certain conditions.
In the context of hydrogen dissociation, the equilibrium constant is determined by the ratio of the concentration—or in the case of gases, the partial pressures—of the products to the reactants, raised to the power of their stoichiometric coefficients. For the dissociation of hydrogen:\[\[\begin{align*}K &= \frac{P_{\mathrm{H}}^2}{P_{\mathrm{H}_2}}\end{align*}\]\]Here, K is directly influenced by temperature, and thus, it varies as the system's temperature changes. The Van't Hoff equation, which relates K with temperature, is used to calculate its value at a specified temperature, such as 3800K in our problem.
In the context of hydrogen dissociation, the equilibrium constant is determined by the ratio of the concentration—or in the case of gases, the partial pressures—of the products to the reactants, raised to the power of their stoichiometric coefficients. For the dissociation of hydrogen:\[\[\begin{align*}K &= \frac{P_{\mathrm{H}}^2}{P_{\mathrm{H}_2}}\end{align*}\]\]Here, K is directly influenced by temperature, and thus, it varies as the system's temperature changes. The Van't Hoff equation, which relates K with temperature, is used to calculate its value at a specified temperature, such as 3800K in our problem.
Partial Pressure
Partial pressure plays a pivotal role in the analysis of gaseous equilibrium. It refers to the pressure that a single gas component would exert if it alone occupied the entire volume of the mixture at the same temperature.
In a mixture, each gas behaves independently, which allows us to calculate its individual partial pressure. The importance of partial pressure becomes apparent when dealing with equilibrium reactions involving gases. For hydrogen's dissociation reaction, the partial pressure of hydrogen (\[\[\begin{align*}P_{\mathrm{H}_2}\end{align*}\]\]) and the dissociated hydrogen atoms (\[\[\begin{align*}P_{\mathrm{H}}\end{align*}\]\]) makes it possible to express the progress of the reaction quantitatively through the equilibrium constant.
In a mixture, each gas behaves independently, which allows us to calculate its individual partial pressure. The importance of partial pressure becomes apparent when dealing with equilibrium reactions involving gases. For hydrogen's dissociation reaction, the partial pressure of hydrogen (\[\[\begin{align*}P_{\mathrm{H}_2}\end{align*}\]\]) and the dissociated hydrogen atoms (\[\[\begin{align*}P_{\mathrm{H}}\end{align*}\]\]) makes it possible to express the progress of the reaction quantitatively through the equilibrium constant.
Degree of Dissociation
The degree of dissociation, represented by the symbol \[\[\begin{align*}\alpha\end{align*}\]\], signifies the fraction of a substance that has dissociated into ions or simpler molecules. This dimensionless quantity is particularly insightful because it indicates the extent of a reaction at equilibrium.
For the dissociation of hydrogen into atomic hydrogen, \[\[\begin{align*}\alpha\end{align*}\]\] tells us the proportion of hydrogen molecules (\[\[\begin{align*}\mathrm{H}_2\end{align*}\]\]) that have been converted to atomic hydrogen (H). It factors in the initial pressure (\[\[\begin{align*}P_0\end{align*}\]\]) and helps us classify the reaction as 'complete' or 'incomplete'. By applying \[\[\begin{align*}\alpha\end{align*}\]\] in the calculation of partial pressures, we conclude the molar fraction of the dissociated molecules and thus determine the effectiveness of the dissociation process.
For the dissociation of hydrogen into atomic hydrogen, \[\[\begin{align*}\alpha\end{align*}\]\] tells us the proportion of hydrogen molecules (\[\[\begin{align*}\mathrm{H}_2\end{align*}\]\]) that have been converted to atomic hydrogen (H). It factors in the initial pressure (\[\[\begin{align*}P_0\end{align*}\]\]) and helps us classify the reaction as 'complete' or 'incomplete'. By applying \[\[\begin{align*}\alpha\end{align*}\]\] in the calculation of partial pressures, we conclude the molar fraction of the dissociated molecules and thus determine the effectiveness of the dissociation process.
Van't Hoff Equation
The Van't Hoff equation is a powerful tool in the field of thermodynamics, expressing the relationship between the equilibrium constant (K) and temperature (T). It links chemical thermodynamics to reaction kinetics and provides insight into how equilibrium shifts with temperature changes.
Mathematically, for a given reaction, the Van't Hoff equation is expressed as:\[\[\begin{align*}\ln K = \frac{-\Delta H^\circ}{R} \left(\frac{1}{T} - \frac{1}{T^\circ}\right)\end{align*}\]\]where \[\[\begin{align*}\Delta H^\circ\end{align*}\]\] is the standard enthalpy change of the reaction, R is the universal gas constant, and \[\[\begin{align*}T^\circ\end{align*}\]\] is the reference temperature. For hydrogen's dissociation, this relationship helps us deduce how the proportion of disassociated hydrogen will vary with an increase in temperature, clearly demonstrating the delicate balance of dynamic chemical equilibriums.
Mathematically, for a given reaction, the Van't Hoff equation is expressed as:\[\[\begin{align*}\ln K = \frac{-\Delta H^\circ}{R} \left(\frac{1}{T} - \frac{1}{T^\circ}\right)\end{align*}\]\]where \[\[\begin{align*}\Delta H^\circ\end{align*}\]\] is the standard enthalpy change of the reaction, R is the universal gas constant, and \[\[\begin{align*}T^\circ\end{align*}\]\] is the reference temperature. For hydrogen's dissociation, this relationship helps us deduce how the proportion of disassociated hydrogen will vary with an increase in temperature, clearly demonstrating the delicate balance of dynamic chemical equilibriums.
