Question

The enthalpy of vaporization of ethanol is 38.7 kJ/mol at its boiling point \(\left(78^{\circ} \mathrm{C}\right) .\) Determine \(\Delta S_{\mathrm{sys}}, \Delta S_{\mathrm{surr}},\) and \(\Delta S_{\mathrm{univ}}\) when 1.00 mole of ethanol is vaporized at \(78^{\circ} \mathrm{C}\) and 1.00 atm.

Short answer

In conclusion, when 1 mole of ethanol is vaporized at 78°C and 1 atm, the change in entropy of the system (ΔS_sys) is \(110.16 J/mol\cdot K\), the change in entropy of the surroundings (ΔS_surr) is \(-110.16 J/mol\cdot K\), and the change in entropy of the universe (ΔS_univ) is \(0 J/mol\cdot K\).

Step-by-step solution

Convert enthalpy of vaporization to J/mol

We are given that the enthalpy of vaporization of ethanol is 38.7 kJ/mol. To convert this value to J/mol, we will multiply by 1000: 38.7 kJ/mol * 1000 J/kJ = 38,700 J/mol Now, the enthalpy of vaporization of ethanol is 38,700 J/mol.

Calculate ΔS_sys

To calculate the change in entropy of the system (ΔS_sys), we will use the formula: ΔS_sys = ΔHvap/T First, we need to convert the boiling point temperature from Celsius to Kelvin by adding 273.15: T = 78°C + 273.15 = 351.15 K Now, we can calculate ΔS_sys: ΔS_sys= (38,700 J/mol) / (351.15 K)= 110.16 J/mol·K The change in entropy of the system is 110.16 J/mol·K.

Calculate ΔS_surr

To calculate the change in entropy of the surroundings (ΔS_surr), we will use the following formula: ΔS_surr = -ΔHvap/T Using the previously calculated values for ΔHvap (38,700 J/mol) and T(351.15 K): ΔS_surr = -(38,700 J/mol) / (351.15 K) = -110.16 J/mol·K The change in entropy of the surroundings is -110.16 J/mol·K.

Calculate ΔS_univ

To calculate the change in entropy of the universe (ΔS_univ), we will add ΔS_sys and ΔS_surr: ΔS_univ = ΔS_sys + ΔS_surr = 110.16 J/mol·K + (-110.16 J/mol·K) = 0 J/mol·K The change in entropy of the universe is 0 J/mol·K. In conclusion, the change in entropy of the system (ΔS_sys) is 110.16 J/mol·K, the change in entropy of the surroundings (ΔS_surr) is -110.16 J/mol·K, and the change in entropy of the universe (ΔS_univ) is 0 J/mol·K when 1 mole of ethanol is vaporized at 78°C and 1 atm.

Key concepts

Entropy Change

Entropy is a fascinating concept in thermodynamics. It helps determine the feasibility of a process by gauging disorder or randomness. When calculating entropy change, we're usually looking at changes during physical transformations. For instance, when ethanol vaporizes, its molecules move from a liquid state (with restricted movement) to a gaseous state (with more freedom to move). This increases system entropy because gaseous molecules are more disordered than liquid molecules.

The change in entropy (\(\Delta S_{\mathrm{sys}}\) is calculated using the enthalpy of vaporization (\(\Delta H_{\text{vap}}\)) and the temperature in Kelvin (\(T\)). The formula is:

• \(\Delta S_{\text{sys}} = \frac{\Delta H_{\text{vap}}}{T}\)

This formula indicates that the entropy change is directly proportional to the heat transferred and inversely proportional to the temperature. For ethanol, vaporizing at its boiling point shows us how thermal energy changes the degree of randomness within the system.

System and Surroundings

In thermodynamics, the universe is often divided into the system and its surroundings. The “system” is the part we are focused on, while the "surroundings" include everything else. When dealing with problems like calculating entropy during a phase change, it's crucial to distinguish between these two categories.

The system here refers to the ethanol undergoing vaporization. Its entropy change (\(\Delta S_{\text{sys}}\)) gives a positive value as the molecules gain freedom and disorder. On the other hand, the surroundings experience an opposite, but equal change in entropy (\(\Delta S_{\text{surr}}\)). This is because as the ethanol absorbs heat from its surroundings, the available energy in the surroundings decreases resulting in a negative entropy change. The formula used is:

• \(\Delta S_{\text{surr}} = -\frac{\Delta H_{\text{vap}}}{T}\)

The universe's net entropy change, calculated as \(\Delta S_{\text{univ}}\), is the combination of both, thus in a reversible process at equilibrium it results in zero: \(\Delta S_{\text{univ}} = 0\). This highlights how system and surroundings are intricately related.

Temperature Conversion

Temperature conversion is a key step in solving many thermodynamic problems. Different temperature scales can be used in scientific investigations. In chemistry and physics, temperature in Kelvin is often used because it starts from absolute zero.

To convert temperatures from Celsius to Kelvin, simply add 273.15. This conversion is essential because thermodynamic formulas usually require temperatures to be in Kelvin for accuracy and standardization. In our ethanol example, the boiling point is 78°C, which converts to:

• \(78^{\circ} \text{C} + 273.15 = 351.15 \text{K}\)

Always ensure your temperatures are correctly converted to avoid mistakes when calculating entropy or other thermodynamic properties.

Spontaneous Processes

Spontaneity in processes is determined by the overall change in the universe's entropy. A process is spontaneous when it can occur without any outside intervention. For many processes at constant pressure and temperature, we assess spontaneity by calculating the change in the universe's entropy \(\Delta S_{\text{univ}}\).

In our ethanol vaporization example, \(\Delta S_{\text{univ}}\) equals zero (0 J/mol·K), demonstrating it is a reversible process at the boiling point. While \(\Delta S_{\text{sys}}\) and \(\Delta S_{\text{surr}}\) are equal and opposite, they balance out, indicating equilibrium rather than true spontaneity. However, if \(\Delta S_{\text{univ}}\) were positive, it would point to a spontaneous direction, for such a process energy from the surroundings would freely assist in vaporization.

• Positive \(\Delta S_{\text{univ}}\) \(\rightarrow\) spontaneous
• Zero \(\Delta S_{\text{univ}}\) \(\rightarrow\) equilibrium
• Negative \(\Delta S_{\text{univ}}\) \(\rightarrow\) non-spontaneous

The critical takeaway is that positive increase in entropy in the universe indicates that energy naturally disperses, emphasizing the preferred nature of spontaneous actions.