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2. The slope of the melting curve of methane (CH4) is given by dP (0.08446 bar/K1.85). T0.85 dT (yes, the exponents are all correct!) from the triple point to arbitrary temperatures. Using the fact that the temperature and pressure of the triple point are 90.68 K and 0.1174 bars, calculate the melting pressure (in bars) of methane at 150 K. (Hint: You will need to integrate to obtain the correct pressure!)

2. The slope of the melting curve of methane (CH4) is given by dP (0.08446 bar/K1.85). T0.85 dT (yes, the exponents are all correct!) from the triple point to arbitrary temperatures. Using the fact that the temperature and pressure of the triple point are 90.68 K and 0.1174 bars, calculate the melting pressure (in bars) of methane at 150 K. (Hint: You will need to integrate to obtain the correct pressure!)

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ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
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### Thermodynamics and Phase Transitions #### 2. Methane Melting Curve The slope of the melting curve for methane (\( \text{CH}_4 \)) is given by the equation: \[ \frac{dP}{dT} = (0.08446 \, \text{bar/K}^{1.85}) \cdot T^{0.85} \] (Yes, the exponents are all correct!) This equation is extended from the triple point to arbitrary temperatures. Given that the temperature and pressure of the triple point are 90.68 K and 0.1174 bars, respectively, calculate the melting pressure (in bars) of methane at 150 K. *Hint: You will need to integrate to obtain the correct pressure!* #### 3. Allotropes of Carbon Data for two "traditional" solid forms of carbon at 300 K are provided below: | Allotrope | \( \Delta H \) combustion (kJ/mol) | \( S^{\circ} \) (J/mol·K) | Density (g/cm\(^3\)) | |-----------|------------------------------------|-------------------------|---------------------| | Diamond | 395.320 | 2.397 | 3.513 | | Graphite | 393.425 | 5.740 | 2.260 | ##### Questions: (a) Calculate the Gibbs free energy (in kJ/mol) of the transition from graphite to diamond at 1 bar and 300 K. In which direction is the process spontaneous? (b) Estimate the pressure (in bars) at which the two allotropes would be in equilibrium at 300 K. Consider how the Gibbs energy changes with pressure at a given temperature \( T \) using the relation \(\left(\frac{\partial G}{\partial P}\right)_T = V\). Assume that the densities of the two allotropes are independent of pressure (Note: This assumption is actually incorrect in real life!).
Transcribed Image Text:### Thermodynamics and Phase Transitions #### 2. Methane Melting Curve The slope of the melting curve for methane (\( \text{CH}_4 \)) is given by the equation: \[ \frac{dP}{dT} = (0.08446 \, \text{bar/K}^{1.85}) \cdot T^{0.85} \] (Yes, the exponents are all correct!) This equation is extended from the triple point to arbitrary temperatures. Given that the temperature and pressure of the triple point are 90.68 K and 0.1174 bars, respectively, calculate the melting pressure (in bars) of methane at 150 K. *Hint: You will need to integrate to obtain the correct pressure!* #### 3. Allotropes of Carbon Data for two "traditional" solid forms of carbon at 300 K are provided below: | Allotrope | \( \Delta H \) combustion (kJ/mol) | \( S^{\circ} \) (J/mol·K) | Density (g/cm\(^3\)) | |-----------|------------------------------------|-------------------------|---------------------| | Diamond | 395.320 | 2.397 | 3.513 | | Graphite | 393.425 | 5.740 | 2.260 | ##### Questions: (a) Calculate the Gibbs free energy (in kJ/mol) of the transition from graphite to diamond at 1 bar and 300 K. In which direction is the process spontaneous? (b) Estimate the pressure (in bars) at which the two allotropes would be in equilibrium at 300 K. Consider how the Gibbs energy changes with pressure at a given temperature \( T \) using the relation \(\left(\frac{\partial G}{\partial P}\right)_T = V\). Assume that the densities of the two allotropes are independent of pressure (Note: This assumption is actually incorrect in real life!).
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