Question

The sublimation of carbon dioxide at \(-78^{\circ} \mathrm{C}\) is given by: $$ \mathrm{CO}_{2}(s) \longrightarrow \mathrm{CO}_{2}(g) \quad \Delta H_{\mathrm{sub}}=25.2 \mathrm{~kJ} / \mathrm{mol} $$ Calculate \(\Delta S_{\text {sub }}\) when \(84.8 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) sublimes at this temperature.

Short answer

The entropy change for this process is 248.8 J/K.

Step-by-step solution

Understand the Problem

We need to calculate the change in entropy, \( \Delta S_{\text{sub}} \), for the sublimation of 84.8 grams of \( \mathrm{CO}_{2} \) at \(-78^{\circ} \mathrm{C}\) with a known enthalpy change, \( \Delta H_{\mathrm{sub}} = 25.2 \mathrm{~kJ/mol} \).

Convert Mass to Moles

First, calculate the number of moles of \( \mathrm{CO}_{2} \) using its molar mass. The molar mass of \( \mathrm{CO}_{2} \) is \( 44.01 \mathrm{~g/mol} \). Calculate the moles:\[\text{moles of } \mathrm{CO}_{2} = \frac{\text{mass}}{\text{molar mass}} = \frac{84.8 \mathrm{~g}}{44.01 \mathrm{~g/mol}} \approx 1.928 \text{ mol}\]

Calculate \( \Delta S_{\text{sub}} \) Using \( \Delta H_{\text{sub}} \)

The change in entropy, \( \Delta S_{\text{sub}} \), can be calculated using the formula: \[\Delta S_{\text{sub}} = \frac{\Delta H_{\text{sub}}}{T}\]where \( T \) is the temperature in Kelvin. First, convert the temperature from Celsius to Kelvin:\[T = -78^{\circ} C + 273.15 = 195.15 \text{ K}\]Now calculate \( \Delta S_{\text{sub}} \):\[\Delta S_{\text{sub}} = \frac{25.2 \mathrm{~kJ/mol}}{195.15 \text{ K}} = 0.1291 \text{ kJ/(mol K)}\]

Adjust for Given Moles of \( \mathrm{CO}_{2} \)

Multiply the entropy change per mole by the number of moles calculated in Step 2 to find the total entropy change:\[\Delta S_{\text{sub, total}} = 0.1291 \text{ kJ/(mol K)} \times 1.928 \text{ mol} \approx 0.2488 \text{ kJ/K}\]

Convert Units if Necessary

Since entropy is often expressed in \( \text{J/K} \), convert the total change in entropy from \( \text{kJ/K} \) to \( \text{J/K} \): \[\Delta S_{\text{sub, total}} = 0.2488 \text{ kJ/K} \times 1000 = 248.8 \text{ J/K}\]

Key concepts

Entropy Change

When a substance undergoes a phase transition, such as sublimation, it experiences a change in entropy. Entropy is a measure of the disorder or randomness in a system. For the sublimation of carbon dioxide, the process involves the transformation from a solid (ordered state) to a gas (more disordered state). This means that the entropy generally increases as the structure of the molecules becomes less ordered.

To calculate the entropy change (\( \Delta S \) ) during sublimation, you can use the relationship with enthalpy change and temperature:

• \( \Delta S = \frac{\Delta H}{T} \)

Here, \( \Delta H \) is the enthalpy change during sublimation, and \( T \) is the temperature in Kelvin. For carbon dioxide, the sublimation entails a positive entropy change because the gaseous molecules are much more spread out compared to their solid counterparts, which results in a higher degree of randomness or disorder.

Enthalpy Change

The concept of enthalpy change (\( \Delta H \) ) is central to understanding phase transitions like sublimation. Enthalpy change is the amount of energy absorbed or released during a process at constant pressure. In the case of sublimation of carbon dioxide, we are interested in the sublimation enthalpy (\( \Delta H_{\text{sub}} \) ).

For carbon dioxide, the enthalpy change for sublimation is given as \( 25.2 \, \text{kJ/mol} \) . This value represents the energy required to transform one mole of CO₂ from solid directly to gas without passing through the liquid phase. This enthalpy is always positive, signifying that the process of sublimation needs the absorption of energy to overcome the forces holding the molecules in the solid structure. For calculations involving moles, this change helps in estimating the total energy absorbed based on the amount of substance.

Carbon Dioxide

Carbon dioxide (CO₂) is a well-known chemical compound composed of one carbon atom covalently double bonded to two oxygen atoms. It is a key component in the Earth's atmosphere, playing a significant role in the greenhouse effect. At standard atmospheric pressure, CO₂ sublimates at \(-78^{\circ}C\) , which is the temperature of dry ice, transitioning directly from solid to gas.

Understanding the properties of CO₂ at various states is crucial for studying its physical chemistry. When CO₂ sublimes under controlled conditions, it's an opportunity to explore fundamental concepts such as entropy and enthalpy change. The examination of these thermodynamic properties highlights the energetic and disorder transformations occurring during this phase change. Overall, carbon dioxide serves as a practical example in thermodynamics due to its straightforward sublimation behavior under specific conditions.