Question

Isooctane, a minor constituent of gasoline, has a boiling point of \(99.3^{\circ} \mathrm{C}\) and a heat of vaporization of \(37.7 \mathrm{~kJ}\) \(\mathrm{mol}^{-1}\). What is \(\Delta S\) (in J \(\mathrm{mol}^{-1} \mathrm{~K}^{-1}\) ) for the vaporization of \(1 \mathrm{~mol}\) of isooctane?

Short answer

The entropy change (Delta S) for the vaporization of 1 mol of isooctane is 101.21 J /mol /K.

Step-by-step solution

Understand the relation between heat of vaporization and entropy change

The change in entropy (Delta S) for the process of vaporization can be calculated using the heat of vaporization (Delta H_{vap}) and the boiling point temperature (T_{b}) in Kelvin, with the formula Delta S = frac{Delta H_{vap}}{T_{b}}. Hence, we need to first convert the boiling point temperature from °C to K.

Convert the boiling point from °C to K

The boiling point in Kelvin (T_{b,K}) can be found by adding 273.15 to the boiling point in °C: T_{b,K} = 99.3 + 273.15 = 372.45 K.

Calculate the change in entropy (Delta S)

Using the formula Delta S = frac{Delta H_{vap}}{T_{b,K}}, we substitute the values to obtain Delta S; hence, Delta S = frac{37.7 kJ/mol}{372.45 K}. Note that the heat of vaporization should be in J/mol to match the units required for Delta S. Thus convert kJ to J by multiplying by 1000: Delta S = frac{37.7 kJ/mol * 1000 J/kJ}{372.45 K} = frac{37700 J/mol}{372.45 K}.

Perform the division

Divide 37700 J/mol by 372.45 K to get the entropy change in J /mol /K: Delta S = 101.21 J/mol /K.

Key concepts

Heat of Vaporization

Heat of vaporization, often denoted as \( \Delta H_{\text{vap}} \), is a crucial concept in both chemistry and physics. It represents the amount of energy required to turn one mole of a liquid into a gas at a constant temperature, typically the substance’s boiling point. For instance, isooctane, a component of gasoline, requires \( 37.7 \text{ kJ/mol} \) of energy to vaporize at its boiling point. This energy breaks the intermolecular forces holding the liquid phase together, allowing molecules to disperse as a gas.

Understanding \( \Delta H_{\text{vap}} \) is essential, as it provides insights into a substance's thermal properties and helps predict how it will behave in different conditions. In the context of solving problems, students should ensure they work with consistent units, converting kilojoules to joules where necessary to align with the standard SI units for entropy, which is measured in \( \text{J/mol K} \) in thermodynamics.

Boiling Point Conversion

To assess entropy change during vaporization, it's necessary to convert the boiling point from Celsius to Kelvin—a process called boiling point conversion. This adjustment accounts for the difference between the two temperature scales. The Kelvin scale is the SI unit for temperature and starts at absolute zero, making it crucial for thermodynamic calculations.

The conversion involves a simple addition: \( T_{\text{b,K}} = T_{\text{b,°C}} + 273.15 \), which places the boiling temperature in the right context for calculating entropy change. For isooctane with a boiling point of \(99.3^\circ \mathrm{C}\), its boiling point in Kelvin will be \(372.45 \text{ K}\). Students must not overlook this step, as failing to use Kelvin can lead to incorrect calculations in thermodynamic equations.

Thermodynamics

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy. In particular, it describes how thermal energy is converted to and from other forms of energy and how it affects matter. One fundamental concept within thermodynamics is entropy, a measure of the disorder or randomness of a system. During phase changes such as vaporization, entropy increases since gas particles have greater freedom of movement and more disorder than liquid particles.

The calculation of entropy change ( \( \Delta S \) ) during vaporization is an application of the second law of thermodynamics and can be calculated by the formula \( \Delta S = \frac{\Delta H_{\text{vap}}}{T_{\text{b,K}}} \). This formula embodies the direct relationship between entropy change, heat of vaporization, and temperature and offers insights into the energetic changes during the phase transition.

Chemical Energetics

Chemical energetics focuses on understanding the energy changes accompanying chemical reactions and phase changes. It provides the background to calculate quantities like enthalpy (\( \Delta H \)) and entropy (\( \Delta S \) ). These calculations are essential for predicting the spontaneity of a process and how energy exchanges with the surroundings.

The heat of vaporization is one aspect of chemical energetics, indicating the energy required for a substance to overcome intermolecular forces during the transition from liquid to gas. By evaluating the entropy change during vaporization, we gain insight into the distribution of energy throughout a system as it changes phases. The calculation presented for isooctane provides not just a numerical answer but a window into the energetic landscape of a chemical undergoing phase change, an integral part of chemical energetics.