Certainly! Let's break down the problem step by step. --- ## **Problem Restatement** **Calculate the values of \( \Delta Q \), \( \Delta W \), \( \Delta U \), and \( \Delta H \) required to take one mole of solid ethanol from its normal freezing point to one mole of ethanol vapor at its normal boiling point.** --- ## **Step 1: Understand the Process** We're taking **1 mole of solid ethanol**: - From its **normal freezing point** (solid ethanol at its melting temperature) - To **ethanol vapor** at its **normal boiling point** This involves several steps: 1. **Heating solid ethanol** from its freezing point to its melting point (if needed) 2. **Melting (fusion)**: Solid \( \rightarrow \) Liquid 3. **Heating liquid ethanol** from melting point to boiling point 4. **Vaporization**: Liquid \( \rightarrow \) Vapor --- ## **Step 2: Required Thermophysical Data** We'll need: - **Freezing point (melting point)** of ethanol: \( T_f = 159 \, \text{K} \) - **Boiling point** of ethanol: \( T_b = 351 \, \text{K} \) - **Molar heat capacity of solid ethanol (\(C_{p,s}\))**: ~\( 112 \, \text{J/mol·K} \) - **Molar heat capacity of liquid ethanol (\(C_{p,l}\))**: ~\( 112 \, \text{J/mol·K} \) - **Enthalpy of fusion (\( \Delta H_{fus} \))**: \( 4.9 \, \text{kJ/mol} \) - **Enthalpy of vaporization (\( \Delta H_{vap} \))**: \( 38.6 \, \text{kJ/mol} \) --- ## **Step 3: Calculating \( \Delta Q \) (Total Heat)** The **total heat required (\( \Delta Q \))** is the sum of: 1. Heat to melt the solid: \( \Delta H_{fus} \) 2. Heat to warm the liquid from \( T_f \) to \( T_b \): \( nC_{p,l}(T_b - T_f) \) 3. Heat to vaporize: \( \Delta H_{vap} \) So, \[ \Delta Q = \Delta H_{fus} + nC_{p,l}(T_b - T_f) + \Delta H_{vap} \] Where \( n = 1 \) mole. --- ## **Step 4: Calculating \( \Delta H \) (Enthalpy Change)** For this process at constant pressure (assuming atmospheric), the total heat for a reversible process equals the enthalpy change: \[ \Delta H = \Delta Q \] --- ## **Step 5: Calculating \( \Delta U \) (Internal Energy Change)** \[ \Delta U = \Delta H - \Delta (PV) \] For ideal gases, \( \Delta (PV) = nR(T_{final} - T_{initial}) \) - \( T_{final} = \) boiling point in K = 351 K - \( T_{initial} = \) freezing point in K = 159 K --- ## **Step 6: Calculating \( \Delta W \) (Work Done)** At constant pressure, the work done by the system is: \[ \Delta W = P \Delta V \] - Most of the volume change occurs during vaporization. --- ## **Step 7: Plug in Values** ### **Data:** - \( T_f = 159 \) K - \( T_b = 351 \) K - \( C_{p,l} = 112 \) J/mol·K - \( \Delta H_{fus} = 4.9 \) kJ/mol = 490 J/mol - \( \Delta H_{vap} = 38.6 \) kJ/mol = 38600 J/mol - \( R = 8.314 \) J/mol·K ### **Calculations:** #### **1. Heating Liquid Ethanol:** \[ q_1 = n C_{p,l} (T_b - T_f) = 1 \times 112 \times (351 - 159) = 1 \times 112 \times 192 = 21,504 \text{ J} \] #### **2. Melting:** \[ q_2 = \Delta H_{fus} = 4,900 \text{ J} \] #### **3. Vaporization:** \[ q_3 = \Delta H_{vap} = 38,600 \text{ J} \] #### **Total Heat (\( \Delta Q \) and \( \Delta H \)):** \[ \Delta Q = \Delta H = 4,900 + 21,504 + 38,600 = 65,004 \text{ J} \] #### **4. Internal Energy (\( \Delta U \)):** \[ \Delta U = \Delta H - nR(T_b - T_f) \] \[ = 65,004 - (1 \times 8.314 \times (351 - 159)) \] \[ = 65,004 - (8.314 \times 192) \] \[ = 65,004 - 1,596.29 = 63,407.71 \text{ J} \] #### **5. Work Done (\( \Delta W \)):** Work is mainly from liquid to vapor (since solids and liquids have much smaller volume changes): \[ \Delta W = P \Delta V \approx nRT_b \] \[ = 1 \times 8.314 \times 351 = 2,918 \text{ J} \] --- ## **Summary Table** | Quantity | Value (J) | |--------------------|-------------| | \( \Delta Q \) | 65,004 | | \( \Delta H \) | 65,004 | | \( \Delta U \) | 63,408 | | \( \Delta W \) | 2,918 | --- ### **Key Points** - **\( \Delta Q \)** and **\( \Delta H \)** are practically the same at constant pressure. - **\( \Delta U \)** is slightly less due to work done against the atmosphere. - **\( \Delta W \)** comes primarily from vaporization. --- #### If you'd like, I can show the step-by-step math for any part, or help you look up any of the data! Let me know what you need more details on.