Problem 6 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Sulfur melts at a temperature of \(115.2^{\circ} \mathrm{C}\) under a pressure of 1 atm. In one process for its mining, sulfur is melted and pumped from underground deposits as a liquid under a pressure of 6 bar Use the Clapeyron equation to estimate the lowest temperature at which sulfur is a liquid at this pressure. (Section 17.2 ) (The enthalpy change of fusion at \(115.2^{\circ} \mathrm{C}\) is \(53.67 \mathrm{Jg}^{-1}\) and the densities of solid and liquid sulfur are \(2.15 \mathrm{gcm}^{-3}\) and \(1.811 \mathrm{gcm}^{-3},\) respectively.

Step-by-Step Solution

Answer

The lowest temperature at which sulfur is a liquid at 6 bar is approximately 119°C.

Step 1: Understand the Clapeyron Equation

The Clapeyron equation is used to determine the relationship between pressure, temperature, and phase changes. It is given by \( \frac{dP}{dT} = \frac{\Delta H_{fusion}}{T \Delta V} \), where \( \Delta H_{fusion} \) is the enthalpy change of fusion, \( T \) is the temperature in Kelvin, and \( \Delta V \) is the change in molar volume between the two phases.

Step 2: Calculate Change in Volume (ΔV)

First, convert the densities of solid and liquid sulfur from \( \mathrm{g/cm^3} \) to \( \mathrm{kg/m^3} \) for consistency with SI units. \( 2.15 \mathrm{g/cm^3} = 2150 \mathrm{kg/m^3} \) and \( 1.811 \mathrm{g/cm^3} = 1811 \mathrm{kg/m^3} \). Use the densities to find the molar volumes: \( V_m = \frac{M}{\rho} \), where \( M = 32.07 \mathrm{g/mol} \) (molar mass of sulfur). Calculate \( V_m \) for both phases, then find \( \Delta V = V_m(liq) - V_m(sol) \).

Step 3: Insert Values into Clapeyron Equation

With \( \Delta V \) and \( \Delta H_{fusion} = 53.67 \mathrm{J/g} = 53670 \mathrm{J/kg} \), rearrange the Clapeyron equation to solve for \( \frac{dT}{dP} = \frac{T \Delta V}{\Delta H_{fusion}} \). We are estimating the shift in melting temperature as pressure changes from 1 atm (converted to 0.987 bar) to 6 bar; therefore, \( dP = 6 - 0.987 = 5.013 \mathrm{bar}\). Convert pressure units to \( \mathrm{Pa} \) by multiplying by \(10^5\).

Step 4: Calculate the New Temperature

Now, find \( dT = \frac{T \Delta V}{\Delta H_{fusion}} \times dP \). Start with the initial temperature in Kelvin: \( T = 115.2 + 273.15 = 388.35 \mathrm{K} \). Substitute all known values to get \( dT \). The new temperature \( T' = T + dT \) will be the estimated melting temperature at 6 bar.

Step 5: Convert Final Temperature to Celsius

After calculating \( T' \) in Kelvin, convert it back to Celsius: \( T'^\circ \mathrm{C} = T' - 273.15 \). This will provide the lowest temperature at which sulfur remains a liquid at 6 bar pressure.

Key Concepts

Sulfur Phase Change

Sulfur is a fascinating element with various allotropes, but at the heart of our focus is the simple phase change from solid to liquid. This transformation occurs at a melting point temperature of approximately 115.2°C under standard atmospheric pressure (1 atm).
In practical applications, like sulfur mining, sulfur is often found in underground deposits. To retrieve it, the sulfur is sometimes melted and extracted in liquid form. This requires understanding the phase change under different pressures. The Clapeyron equation provides a tool to predict the temperature at which sulfur will change phases when the pressure is altered.
During the phase change, the structure of sulfur molecules rearranges. The molecules in the solid phase vibrate with less freedom compared to the more fluid and disordered arrangement in the liquid phase. So, understanding the sulfur phase change is essential in chemical processes where pressure and temperature control are necessary.

Enthalpy Change of Fusion

The enthalpy change of fusion is an important thermodynamic quantity. It represents the heat required to change a substance from a solid to a liquid at its melting point. For sulfur, this value is 53.67 J/g at 115.2°C. This energy aids in breaking the intermolecular attractions in the solid phase, allowing sulfur molecules to flow more freely in the liquid phase.
In sulfur mining, knowing the enthalpy change of fusion helps in energy management. It tells us how much energy is needed to maintain sulfur in a liquid state at the required conditions. It also plays a key role in the Clapeyron equation, which uses this energy value for calculations regarding phase changes under varied pressures.
Thus, enthalpy change of fusion is crucial for calculating the new conditions required for sulfur extraction at different pressures.

Phase Transitions

Phase transitions reflect the change of matter from one state to another, like solids to liquids or liquids to gases. For sulfur, the transition from solid to liquid is most relevant in mining operations. Phase transitions occur through the addition or removal of energy. The energy needed varies depending on the current state and desired state.
The Clapeyron equation becomes useful here as it models how pressure and temperature shifts affect the equilibrium between different phases. When sulfur is subjected to increased pressure, as in underground deposits at 6 bar, it necessitates recalculating its melting point.
Understanding these transitions helps optimize processes where precise temperature and pressure balance is required to maintain a specific phase, ultimately yielding practical applications such as efficient sulfur extraction.

Molar Volume Change

Molar volume refers to the volume occupied by one mole of a substance, and it plays a significant role during phase changes. For sulfur, we calculate molar volumes using the densities of both solid and liquid phases. With sulfur's density: 2.15 g/cm³ for solid and 1.811 g/cm³ for liquid, we first need to convert these values into SI units.
Once expressed in kg/m³, the densities are used alongside the molar mass of sulfur (32.07 g/mol) to determine molar volumes. The difference in molar volumes between the liquid and solid states, denoted as \( \Delta V \), is critical for the Clapeyron equation.
It describes how much the volume changes during melting. This volume change impacts how pressure affects the melting point, thus, it’s paramount for calculating new phase transition temperatures under various pressures. Molar volume change is a key factor in thermodynamics, specifically when working with the equilibrium of phases in different conditions.