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Q11.21(a):
Answer the following questions based on the P-T phase diagram of carbon dioxide: (a) At what temperature and pressure can the solid, liquid and vapour phases of $CO_2$ co-exist in equilibrium ?
Solution :
Step 1: Theoretical Basis of Phase Equilibrium
The state of a substance where three phases (solid, liquid, and gas/vapour) exist simultaneously in thermodynamic equilibrium is defined as the Triple Point. [Per the Gibbs Phase Rule, $F = C - P + 2$, where $F$ is degrees of freedom, $C$ is components (1 for $CO_2$), and $P$ is phases (3). At the triple point, $F = 1 - 3 + 2 = 0$, meaning the point is invariant and fixed at a specific temperature and pressure].
Step 2: Empirical Data for Carbon Dioxide
For Carbon Dioxide ($CO_2$), the triple point parameters are experimentally determined constants within the $P-T$ phase diagram. Unlike water, the triple point of $CO_2$ occurs at a pressure significantly higher than standard atmospheric pressure ($1 \text{ atm} \approx 1.013 \times 10^5 \text{ Pa}$).
| Parameter | Value |
|---|---|
| Temperature ($T_{tp}$) | $216.55 \text{ K}$ (or $-56.6^\circ\text{C}$) |
| Pressure ($P_{tp}$) | $5.11 \text{ atm}$ (or $5.17 \times 10^5 \text{ Pa}$) |
Step 3: Phase Diagram Representation
The following SVG illustrates the $P-T$ phase diagram for $CO_2$, highlighting the triple point where the sublimation, fusion, and vaporisation curves intersect.
Step 4: Analytical Conclusion
Because the triple point pressure ($5.11 \text{ atm}$) is greater than $1 \text{ atm}$, solid $CO_2$ (dry ice) at standard atmospheric pressure cannot transition into a liquid phase when heated. Instead, it undergoes sublimation directly from a solid to a gaseous state, as the $1 \text{ atm}$ isobar lies entirely below the triple point on the $P-T$ diagram. [Based on the Clausius-Clapeyron relation describing phase boundaries].
Final Solution: The solid, liquid, and vapour phases of $CO_2$ co-exist in equilibrium at a temperature of $216.55 \text{ K}$ and a pressure of $5.11 \text{ atm}$.
More Questions from Class 11 Physics Thermal Properties of Matter ADDITIONAL EXERCISES
- Q11.21(b): Answer the following questions based on the P-T phase diagram of carbon dioxide: (b) What is the effect of decrease of pressure on the fusion and boiling point of $CO_2$ ?
- Q11.21(c): Answer the following questions based on the P-T phase diagram of carbon dioxide: (c) What are the critical temperature and pressure for $CO_2$ ? What is their significance ?
- Q11.21(d): Answer the following questions based on the P-T phase diagram of carbon dioxide: (d) Is $CO_2$ solid, liquid or gas at (a) –70 °C under 1 atm, (b) –60 °C under 10 atm, (c) 15 °C under 56 atm ?
- Q11.22(a): Answer the following questions based on the P – T phase diagram of $CO_2$: (a) $CO_2$ at 1 atm pressure and temperature – 60 °C is compressed isothermally. Does it go through a liquid phase ?
- Q11.22(b): Answer the following questions based on the P – T phase diagram of $CO_2$: (b) What happens when $CO_2$ at 4 atm pressure is cooled from room temperature at constant pressure ?
- Q11.22(c): Answer the following questions based on the P – T phase diagram of $CO_2$: (c) Describe qualitatively the changes in a given mass of solid $CO_2$ at 10 atm pressure and temperature –65 °C as it is heated up to room temperature at constant pressure.
- Q11.22(d): Answer the following questions based on the P – T phase diagram of $CO_2$: (d) $CO_2$ is heated to a temperature 70 °C and compressed isothermally. What changes in its properties do you expect to observe ?
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