substance x is known to exist at 1 atm in the solid, liquid, or vapor phase, depending on the temperature. additionally, the values of these other properties of x have been determined:
- melting point: 30. °c; enthalpy of fusion: 5.00 kj/mol; density: 1.90 g/cm³ (solid), 1.60 g/ml (liquid)
- boiling point: 85. °c; enthalpy of vaporization: 27.00 kj/mol; heat capacity: 41. j·k⁻¹·mol⁻¹ (solid), 78. j·k⁻¹·mol⁻¹ (liquid), 77. j·k⁻¹·mol⁻¹ (vapor)
you may also assume x behaves as an ideal gas in the vapor phase.
suppose a small sample of x at 20 °c is put into an evacuated flask and heated at a constant rate until 15.0 kj/mol of heat has been added to the sample.
graph the temperature of the sample that would be observed during this experiment.
graph with “temperature (°c)” axis (ranging ~30–120 °c) and grid lines provided

Question

Accepted Answer

To solve this problem, we analyze the heating process of Substance X step by step:

### Step 1: Heat Solid X from $ 20^\circ \text{C} $ to Melting Point ($ 30^\circ \text{C} $)
- **Phase**: Solid
- **Heat Capacity ($ C_{\text{solid}} $)**: $ 41 \, \text{J·K}^{-1}·\text{mol}^{-1} $ (note: $ \Delta T $ in K = $ \Delta T $ in $ ^\circ \text{C} $)
- $ \Delta T = 30 - 20 = 10 \, \text{K} $
- Heat required ($ q_1 $):
  $$
  q_1 = n \cdot C_{\text{solid}} \cdot \Delta T
  $$
  Assume $ n = 1 \, \text{mol} $ (since heat is given per mole):
  $$
  q_1 = 1 \cdot 41 \cdot 10 = 410 \, \text{J} = 0.41 \, \text{kJ}
  $$

### Step 2: Melt Solid X to Liquid (Phase Change at $ 30^\circ \text{C} $)
- **Enthalpy of Fusion ($ \Delta H_{\text{fusion}} $)**: $ 5.00 \, \text{kJ/mol} $
- Heat required ($ q_2 $):
  $$
  q_2 = n \cdot \Delta H_{\text{fusion}} = 1 \cdot 5.00 = 5.00 \, \text{kJ}
  $$

### Step 3: Heat Liquid X from $ 30^\circ \text{C} $ to Boiling Point ($ 85^\circ \text{C} $)
- **Phase**: Liquid
- **Heat Capacity ($ C_{\text{liquid}} $)**: $ 78 \, \text{J·K}^{-1}·\text{mol}^{-1} $
- $ \Delta T = 85 - 30 = 55 \, \text{K} $
- Heat required ($ q_3 $):
  $$
  q_3 = n \cdot C_{\text{liquid}} \cdot \Delta T
  $$
  $$
  q_3 = 1 \cdot 78 \cdot 55 = 4290 \, \text{J} = 4.29 \, \text{kJ}
  $$

### Step 4: Boil Liquid X to Vapor (Phase Change at $ 85^\circ \text{C} $)
- **Enthalpy of Vaporization ($ \Delta H_{\text{vaporization}} $)**: $ 27.00 \, \text{kJ/mol} $
- Heat required ($ q_4 $):
  $$
  q_4 = n \cdot \Delta H_{\text{vaporization}} = 1 \cdot 27.00 = 27.00 \, \text{kJ}
  $$

### Step 5: Heat Vapor X (If Remaining Heat Exists)
Total heat applied: $ 15.0 \, \text{kJ} $ (per mole).
Sum of $ q_1 + q_2 + q_3 = 0.41 + 5.00 + 4.29 = 9.70 \, \text{kJ} $.
Remaining heat: $ 15.0 - 9.70 = 5.30 \, \text{kJ} $, which is less than $ q_4 = 27.00 \, \text{kJ} $. Thus, we only partially heat the vapor.

- **Phase**: Vapor
- **Heat Capacity ($ C_{\text{vapor}} $)**: $ 77 \, \text{J·K}^{-1}·\text{mol}^{-1} = 0.077 \, \text{kJ·K}^{-1}·\text{mol}^{-1} $
- Heat available for vapor heating: $ q_5 = 5.30 \, \text{kJ} $
- Temperature change of vapor ($ \Delta T_{\text{vapor}} $):
  $$
  \Delta T_{\text{vapor}} = \frac{q_5}{n \cdot C_{\text{vapor}}} = \frac{5.30}{1 \cdot 0.077} \approx 68.8 \, \text{K}
  $$
- Final temperature of vapor: $ 85 + 68.8 \approx 153.8^\circ \text{C} $

### Graphing the Temperature vs. Heat
1. **Solid Heating**: Linear increase from $ 20^\circ \text{C} $ to $ 30^\circ \text{C} $ (slope $ \propto 1/C_{\text{solid}} $).
2. **Melting (Phase Change)**: Horizontal line at $ 30^\circ \text{C} $ (constant temperature, heat used for fusion).
3. **Liquid Heating**: Linear increase from $ 30^\circ \text{C} $ to $ 85^\circ \text{C} $ (slope $ \propto 1/C_{\text{liquid}} $).
4. **Boiling (Phase Change)**: Horizontal line at $ 85^\circ \text{C} $ (constant temperature, heat used for vaporization).
5. **Vapor Heating**: Linear increase from $ 85^\circ \text{C} $ to $ \approx 154^\circ \text{C} $ (slope $ \propto 1/C_{\text{vapor}} $).

### Final Temperature
After $ 15.0 \, \text{kJ/mol} $ of heat, the sample is in the vapor phase at approximately $ \boldsymbol{154^\circ \text{C}} $ (or more precisely, $ 85 + 68.8 \approx 153.8^\circ \text{C} $).

(Note: The graph should plot temperature on the y-axis and heat added on the x-axis, with horizontal segments at $ 30^\circ \text{C} $ (melting) and $ 85^\circ \text{C} $ (boiling), and linear segments between them with slopes inversely proportional to the heat capacities of each phase.)

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QUESTION IMAGE

Question

Explanation:

Step 1: Heat Solid X from \( 20^\circ \text{C} \) to Melting Point (\( 30^\circ \text{C} \))

Step 2: Melt Solid X to Liquid (Phase Change at \( 30^\circ \text{C} \))

Step 3: Heat Liquid X from \( 30^\circ \text{C} \) to Boiling Point (\( 85^\circ \text{C} \))

Step 4: Boil Liquid X to Vapor (Phase Change at \( 85^\circ \text{C} \))

Step 5: Heat Vapor X (If Remaining Heat Exists)

Graphing the Temperature vs. Heat

Final Temperature

Answer: