The normal boiling point of bromine is 58.8 °C, and the standard entropies of the liquid and vapor are S°[Br2(l) = 152.2 J/(K*mol); S°[Br2(g) = 245.4 J/(K*mol). At what temperature does bromine have a vapor pressure of 227 mmHg?

Verified step by step guidance

1. Convert the given vapor pressure from mmHg to atm using the conversion factor (1 atm = 760 mmHg).
2. Use the Clausius-Clapeyron equation to relate the change in vapor pressure with temperature. The equation is: \(\ln\left(\frac{P_2}{P_1}\right) = \frac{\Delta H_{vap}}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)\), where \(P_1\) and \(P_2\) are the initial and final pressures, \(T_1\) and \(T_2\) are the initial and final temperatures, \(\Delta H_{vap}\) is the enthalpy of vaporization, and R is the gas constant.
3. Calculate the enthalpy of vaporization (\(\Delta H_{vap}\)) using the entropy values at the boiling point. The relationship is \(\Delta H_{vap} = T_{boil} \times (S°[Br2(g)] - S°[Br2(l)])\), where \(T_{boil}\) is the boiling temperature in Kelvin.
4. Substitute the known values into the Clausius-Clapeyron equation. Use the boiling point of bromine as \(T_1\) and convert it to Kelvin by adding 273.15 to the Celsius value. Solve for \(T_2\) in Kelvin.
5. Convert the final temperature \(T_2\) from Kelvin back to Celsius by subtracting 273.15.

Key Concepts

Boiling Point and Vapor Pressure

The boiling point of a substance is the temperature at which its vapor pressure equals the external pressure surrounding the liquid. For bromine, the normal boiling point is 58.8 °C at 1 atm pressure. When the vapor pressure of a liquid reaches a specific value, such as 227 mmHg, it indicates the temperature at which the liquid will boil under that pressure.

Entropy and Phase Changes

Entropy is a measure of the disorder or randomness in a system. In phase changes, such as from liquid to gas, the entropy of the gas is typically higher than that of the liquid due to increased molecular motion. The standard entropies provided for liquid and vapor bromine (152.2 J/(K*mol) and 245.4 J/(K*mol), respectively) are essential for calculating the temperature at which the vapor pressure reaches 227 mmHg.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation relates the change in vapor pressure with temperature to the enthalpy of vaporization. It can be used to determine the temperature at which a specific vapor pressure occurs by incorporating the entropies of the liquid and vapor phases. This equation is crucial for solving the problem of finding the temperature at which bromine has a vapor pressure of 227 mmHg.