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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.
Solution :
Initial State and Thermodynamic Coordinates
Given the initial state of the carbon dioxide ($CO_2$) sample:
- Pressure ($P$): $10\text{ atm}$
- Temperature ($T$): $-65\text{ °C}$
- Phase: Solid ($CO_2$ ice/dry ice)
Phase Transition Analysis
As the system is heated at a constant pressure of $10\text{ atm}$, it traverses the following path in the $P-T$ coordinate plane:
- Heating the Solid: The temperature increases from $-65\text{ °C}$. The solid $CO_2$ undergoes sensible heating, increasing in internal energy without a phase change until the fusion curve is intersected.
- Melting (Solid to Liquid): At $10\text{ atm}$, the melting point of $CO_2$ is approximately $-56\text{ °C}$. Upon reaching this temperature, the solid absorbs latent heat of fusion [Per the First Law of Thermodynamics, $dQ = dU + PdV$]. The phase transition from solid to liquid occurs while the temperature remains constant during the phase change.
- Heating the Liquid: Once the mass is entirely in the liquid phase, further heating causes the temperature of the liquid $CO_2$ to rise.
- Vaporization (Liquid to Gas): As heating continues toward room temperature ($20\text{ °C}$–$25\text{ °C}$), the system approaches the boiling point. At $10\text{ atm}$, which is lower than the critical pressure of $CO_2$ ($73\text{ atm}$), the liquid will eventually cross the vaporization curve and undergo a phase change into the gaseous state (vapor).
Visual Representation of the Phase Transition Path
Summary of Thermodynamic Evolution
The progression of the system can be summarized by the following table:
| Phase | Temperature Range | Mechanism |
|---|---|---|
| Solid | $-65\text{ °C}$ to $\approx -56\text{ °C}$ | Sensible heating of solid |
| Solid + Liquid | $\approx -56\text{ °C}$ | Latent heat of fusion |
| Liquid | $-56\text{ °C}$ to $T_{boiling}$ | Sensible heating of liquid |
| Gas | $T_{boiling}$ to Room Temp | Expansion/Vaporization |
Final Solution: The solid $CO_2$ first undergoes sensible heating, then melts into liquid $CO_2$ at approximately $-56\text{ °C}$, and finally, as heating continues toward room temperature at $10\text{ atm}$, the liquid will eventually vaporize into gas.
More Questions from Class 11 Physics Thermal Properties of Matter ADDITIONAL EXERCISES
- 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 ?
- 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(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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