Question

Consider the decomposition of calcium carbonate. $$ \mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$$ Calculate the pressure in atm of \(\mathrm{CO}_{2}\) in an equilibrium process (a) at \(25^{\circ} \mathrm{C}\) and (b) at \(800^{\circ} \mathrm{C}\). Assume that \(\Delta H^{\circ}=177.8 \mathrm{~kJ} / \mathrm{mol}\) and \(\Delta S^{\circ}=160.5 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\) for the temperature range.

Short answer

The pressure of CO2 in an equilibrium process will be equal to \(K_1\) at \(25^{\circ}C\) and \(K_2\) at \(800^{\circ}C\). The exact values for \(K_1\) and \(K_2\) will depend on the specific numerical calculations in the steps outlined above. Use these steps to find the specific values.

Step-by-step solution

Transform the temperatures

First, the given temperatures should be converted from Celsius to Kelvin, because thermodynamic calculations deal with Kelvin scale. The conversion from Celsius to Kelvin is done with the formula \(K = ^{\circ}C + 273.15\). Therefore, the temperatures in Kelvin are \(T_1 = 25 + 273.15 = 298.15 K\) and \(T_2 = 800 + 273.15 = 1073.15 K\).

Convert \(\Delta H\) and \(\Delta S\) into suitable units

The given \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) need to be in the same units before they can be used in the Van't Hoff equation. Given that \(\Delta H^{\circ} = 177.8 kJ/mol\), it can be converted to \(J/mol\) by multiplying by \(1000\), yielding \(\Delta H^{\circ} = 177800 J/mol\). The given \(\Delta S^{\circ}\) is already in the appropriate units \(J/(K \cdot mol)\) and doesn't need to be converted.

Applying Van't Hoff equation

Now, calculate the value of equilibrium constant \(K_1\) at \(T_1 = 298.15 K\) by applying the Van't Hoff equation, which in logarithmic form is \(\ln K_1 = -\Delta H^{\circ}/(RT_1) + \Delta S^{\circ}/R\). You can use a standard molar gas constant \(R = 8.314 J/(K \cdot mol)\). After doing the calculation, take exponent on both sides to get \(K_1\).

Calculate the pressure at 25C

As the only gaseous component is CO2, at \(T_1\), the pressure of CO2 can be considered as the equilibrium constant, because the pressure units of a gas can be expressed in terms of atmospheres, bar, or torr etc. Therefore, you can take \(P = K_1\) at \(T_1\).

Calculate the value of equilibrium constant \(K_2\) at \(T_2 = 1073.15 K\)

Once again, apply the Van't Hoff equation to find the equilibrium constant at the second temperature. However, this time the equation will look like: \(\ln K_2 = \ln K_1 - \Delta H^{\circ}/R \cdot (1/T_2 - 1/T_1)\). Calculate the logarithm value of \(K_2\) and then take exponent to get the value of \(K_2\), which at this temperature will represent the pressure of CO2.

Calculate the pressure at 800C

Similar to step 4, the equilibrium constant at 800C is equal to the pressure of CO2. Since the only gaseous product is CO2, the total pressure in the system at equilibrium is equal to the partial pressure of CO2 namely \(P = K_2\) at \(T_2\).

Key concepts

Thermodynamics

Thermodynamics is the branch of physical science concerned with heat and temperature and their relation to energy and work. It encompasses the principles that describe how energy is transferred and transformed, and it plays a pivotal role in chemical processes. When dealing with chemical reactions, thermodynamics helps in understanding whether a reaction can occur spontaneously and how energy changes in a system.

Three main laws of thermodynamics are worth noting:

• The First Law, which states that energy cannot be created or destroyed. It only changes form.
• The Second Law, which introduces the concept of entropy, indicating that energy spontaneously spreads from regions of higher concentration to lower concentration.
• The Third Law, suggesting that as temperature approaches absolute zero, entropy approaches a constant minimum.

In the context of chemical equilibrium, thermodynamics can help predict the direction and extent to which a reaction will proceed. By investigating enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)), students can understand not only the feasibility but also the favorability of chemical changes under different conditions.

Van't Hoff Equation

The Van't Hoff equation is a fundamental equation in chemical thermodynamics that relates the change in the equilibrium constant (\( K \)) of a reaction to the change in temperature. It is a crucial tool for understanding how temperature affects chemical reactions, specifically their equilibrium positions.

The equation is given by:
\[\ln(K_2/K_1) = -\frac{\Delta H^{\circ}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\]where:

• \( \Delta H^{\circ} \) is the change in enthalpy of the reaction (in \( J/mol \)).
• \( R \) is the universal gas constant (8.314 \( J/(K \cdot mol) \)).
• \( T_1 \) and \( T_2 \) are the initial and final temperatures (in Kelvin).
• \( K_1 \) and \( K_2 \) are the equilibrium constants at temperatures \( T_1 \) and \( T_2 \) respectively.

This equation indicates that if a reaction is endothermic (\( \Delta H^{\circ} > 0 \)), an increase in temperature will result in an increase in the equilibrium constant \( K \), pushing the equilibrium towards the products. Conversely, for exothermic reactions (\( \Delta H^{\circ} < 0 \)), warming the system will decrease \( K \), favoring the reactants.

Equilibrium Constant

The equilibrium constant (\( K \)) of a chemical reaction is a fundamental concept in chemistry that quantifies the ratio of the concentrations of products to reactants at equilibrium. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, meaning the system is at a balance, and the concentrations of the involved species remain constant.

For a general reaction \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant expression in terms of concentration (\( K_c \)) is:
\[K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]It's crucial to note that \( K \) values are affected by temperature, as indicated by the Van't Hoff equation. Yet, they remain constant at a given temperature irrespective of the starting species concentrations.

In reactions where one of the components is a gas, like the decomposition of calcium carbonate to yield gaseous \( CO_2 \), the equilibrium constant can also be expressed in terms of partial pressures, leading to \( K_p \). The relationship between \( K_c \) and \( K_p \) can be evaluated using the equation: \( K_p = K_c(RT)^{\Delta n} \), where \( \Delta n \) is the change in moles of gas in the reaction. In the exercise, \( K \) is equivalent to the partial pressure of \( CO_2 \), since it is the only gaseous product and directly relates to how much of the decomposition reaction has occurred.