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Q12.6(d):
Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following : (d) Do the intermediate states of the system (before settling to the final equilibrium state) lie on its P-V-T surface ?
Solution :
1. Analysis of System Constraints and Thermodynamics
The system under consideration is a classic example of Joule expansion or free expansion. The core parameters are:
- System: Two cylinders (A and B) connected by a stopcock.
- Initial State: Cylinder A contains an ideal gas at $P_0, V_0, T_0$. Cylinder B is at vacuum ($P=0$).
- Insulation: The system is thermally insulated ($dQ = 0$), implying an adiabatic process.
- Process: The stopcock is opened suddenly, causing the gas to expand into the vacuum of cylinder B.
2. Conceptualizing the P-V-T Surface
The P-V-T surface is a representation of all possible equilibrium states of a thermodynamic system. Any point $(P, V, T)$ residing on the P-V-T surface corresponds to a state where the system's properties are uniform throughout its volume [Per the Postulates of Equilibrium Thermodynamics].
3. Analyzing the Intermediate States
When the stopcock is suddenly opened, the following physical phenomena occur:
- Turbulence and Non-Equilibrium: The gas rushes from cylinder A into cylinder B. During this interval, the gas velocity is high, and the pressure and temperature are non-uniform across the two volumes [Per the Laws of Fluid Dynamics].
- Lack of Uniformity: Because the system is in a state of rapid transition, it is impossible to define a single unique pressure $P$ or temperature $T$ for the entire system.
- Deviation from Equilibrium: Since a state of thermodynamic equilibrium is not maintained during the expansion, the intermediate states do not satisfy the equation of state (e.g., the Ideal Gas Law $PV = nRT$).
4. Conclusion on P-V-T Surface Validity
The P-V-T surface is strictly defined only for states of equilibrium. During the sudden expansion, the system passes through a series of "non-equilibrium states." In these states, the thermodynamic variables ($P, V, T$) are not well-defined for the bulk of the gas because the gas undergoes turbulent, irreversible motion. Therefore, these intermediate states do not represent equilibrium points and cannot lie on the equilibrium P-V-T surface.
Final Solution: No, the intermediate states do not lie on the P-V-T surface because they are states of non-equilibrium where the thermodynamic variables $P$, $V$, and $T$ cannot be uniquely defined for the system as a whole.
More Questions from Class 11 Physics Thermodynamics EXERCISES
- Q5.1: Choose the correct answer. A thermodynamic state function is a quantity
- Q5.10: Calculate the enthalpy change on freezing of 1.0 mol of water at (10.0^{\circ}\text{C}) to ice at (–10.0^{\circ}\text{C}). (\Delta_{\text{fus}}H = 6.03 \text{ kJ mol}^{-1}) at (0^{\circ}\text{C}). (C_{p} [\text{H}_{2}\text{O(l)}] = 75.3 \text{ J mol}^{-1} \text{ K}^{-1}) (C_{p} [\text{H}_{2}\text{O(s)}] = 36.8 \text{ J mol}^{-1} \text{ K}^{-1})
- Q5.11: Enthalpy of combustion of carbon to (\text{CO}_{2}) is (–393.5 \text{ kJ mol}^{-1}). Calculate the heat released upon formation of 35.2 g of (\text{CO}_{2}) from carbon and dioxygen gas.
- Q5.12: Enthalpies of formation of (\text{CO(g)}), (\text{CO}_{2}(\text{g})), (\text{N}_{2}\text{O(g)}) and (\text{N}_{2}\text{O}_{4}(\text{g})) are –110, – 393, 81 and (9.7 \text{ kJ mol}^{-1}) respectively. Find the value of (\Delta_{r}H) for the reaction: (\text{N}_{2}\text{O}_{4}(\text{g}) + 3\text{CO(g)} \rightarrow \text{N}_{2}\text{O(g)} + 3\text{CO}_{2}(\text{g}))
- Q5.13: Given (\text{N}_{2}(\text{g}) + 3\text{H}_{2}(\text{g}) \rightarrow 2\text{NH}_{3}(\text{g}); \Delta_{r}H^{\ominus} = –92.4 \text{ kJ mol}^{-1}) What is the standard enthalpy of formation of (\text{NH}_{3}) gas?
- Q5.14: Calculate the standard enthalpy of formation of (\text{CH}_{3}\text{OH(l)}) from the following data: (\text{CH}_{3}\text{OH (l)} + \frac{3}{2} \text{O}_{2}(\text{g}) \rightarrow \text{CO}_{2}(\text{g}) + 2\text{H}_{2}\text{O(l)}) ; (\Delta_{r}H^{\ominus} = –726 \text{ kJ mol}^{-1}) (\text{C(graphite)} + \text{O}_{2}(\text{g}) \rightarrow \text{CO}_{2}(\text{g})) ; (\Delta_{c}H^{\ominus} = –393 \text{ kJ mol}^{-1}) (\text{H}_{2}(\text{g}) + \frac{1}{2} \text{O}_{2}(\text{g}) \rightarrow \text{H}_{2}\text{O(l)}); (\Delta_{f}H^{\ominus} = –286 \text{ kJ mol}^{-1}).
- Q5.15: Calculate the enthalpy change for the process (\text{CCl}_{4}(\text{g}) \rightarrow \text{C(g)} + 4 \text{Cl(g)}) and calculate bond enthalpy of C – Cl in (\text{CCl}_{4}(\text{g})). (\Delta_{\text{vap}}H^{\ominus}(\text{CCl}_{4}) = 30.5 \text{ kJ mol}^{-1}). (\Delta_{f}H^{\ominus} (\text{CCl}_{4}) = –135.5 \text{ kJ mol}^{-1}). (\Delta_{a}H^{\ominus} (\text{C}) = 715.0 \text{ kJ mol}^{-1}), where (\Delta_{a}H^{\ominus}) is enthalpy of atomisation (\Delta_{a}H^{\ominus} (\text{Cl}_{2}) = 242 \text{ kJ mol}^{-1})
- Q5.16: For an isolated system, (\Delta U = 0), what will be (\Delta S) ?
- Q5.17: For the reaction at 298 K, (2A + B \rightarrow C) (\Delta H = 400 \text{ kJ mol}^{-1}) and (\Delta S = 0.2 \text{ kJ K}^{-1} \text{ mol}^{-1}) At what temperature will the reaction become spontaneous considering (\Delta H) and (\Delta S) to be constant over the temperature range.
- Q5.18: For the reaction, (2 \text{Cl(g)} \rightarrow \text{Cl}_{2}(\text{g})), what are the signs of (\Delta H) and (\Delta S) ?
- Q5.19: For the reaction (2 \text{A(g)} + \text{B(g)} \rightarrow 2\text{D(g)}) (\Delta U^{\ominus} = –10.5 \text{ kJ}) and (\Delta S^{\ominus} = –44.1 \text{ JK}^{-1}). Calculate (\Delta G^{\ominus}) for the reaction, and predict whether the reaction may occur spontaneously.
- Q5.2: For the process to occur under adiabatic conditions, the correct condition is:
- Q5.20: The equilibrium constant for a reaction is 10. What will be the value of (\Delta G^{\ominus}) ? (R = 8.314 \text{ JK}^{-1} \text{ mol}^{-1}), (T = 300 \text{ K}).
- Q5.21: Comment on the thermodynamic stability of (\text{NO(g)}), given (\frac{1}{2} \text{N}_{2}(\text{g}) + \frac{1}{2}\text{O}_{2}(\text{g}) \rightarrow \text{NO(g)}); (\Delta_{r}H^{\ominus} = 90 \text{ kJ mol}^{-1}) (\text{NO(g)} + \frac{1}{2}\text{O}_{2}(\text{g}) \rightarrow \text{NO}_{2}(\text{g}): \Delta_{r}H^{\ominus}= –74 \text{ kJ mol}^{-1})
- Q5.22: Calculate the entropy change in surroundings when 1.00 mol of (\text{H}_{2}\text{O(l)}) is formed under standard conditions. (\Delta_{f}H^{\ominus} = –286 \text{ kJ mol}^{-1}).
- Q5.3: The enthalpies of all elements in their standard states are:
- Q5.4: (\Delta U^{\ominus}) of combustion of methane is (– X \text{ kJ mol}^{-1}). The value of (\Delta H^{\ominus}) is
- Q5.5: The enthalpy of combustion of methane, graphite and dihydrogen at 298 K are, (–890.3 \text{ kJ mol}^{-1}), (–393.5 \text{ kJ mol}^{-1}), and (–285.8 \text{ kJ mol}^{-1}) respectively. Enthalpy of formation of (\text{CH}_{4}(\text{g})) will be
- Q5.6: A reaction, (A + B \rightarrow C + D + q) is found to have a positive entropy change. The reaction will be
- Q5.7: In a process, 701 J of heat is absorbed by a system and 394 J of work is done by the system. What is the change in internal energy for the process?
- Q5.8: The reaction of cyanamide, (\text{NH}_{2}\text{CN (s)}), with dioxygen was carried out in a bomb calorimeter, and (\Delta U) was found to be (–742.7 \text{ kJ mol}^{-1}) at 298 K. Calculate enthalpy change for the reaction at 298 K. (\text{NH}_{2}\text{CN(g)} + \frac{3}{2}\text{O}_{2}(\text{g}) \rightarrow \text{N}_{2}(\text{g}) + \text{CO}_{2}(\text{g}) + \text{H}_{2}\text{O(l)})
- Q5.9: Calculate the number of kJ of heat necessary to raise the temperature of 60.0 g of aluminium from (35^{\circ}\text{C}) to (55^{\circ}\text{C}). Molar heat capacity of Al is (24 \text{ J mol}^{-1} \text{ K}^{-1}).
- Q12.1: A geyser heats water flowing at the rate of 3.0 litres per minute from $27^\circ C$ to $77^\circ C$. If the geyser operates on a gas burner, what is the rate of consumption of the fuel if its heat of combustion is $4.0 \times 10^4 J/g$ ?
- Q12.10: A refrigerator is to maintain eatables kept inside at $9^\circ C$. If room temperature is $36^\circ C$, calculate the coefficient of performance.
- Q12.2: What amount of heat must be supplied to $2.0 \times 10^{-2}$ kg of nitrogen (at room temperature) to raise its temperature by $45^\circ C$ at constant pressure ? (Molecular mass of $N_2 = 28$; $R = 8.3 J mol^{-1} K^{-1}$.)
- Q12.3(a): Explain why (a) Two bodies at different temperatures $T_1$ and $T_2$ if brought in thermal contact do not necessarily settle to the mean temperature $(T_1 + T_2)/2$.
- Q12.3(b): Explain why (b) The coolant in a chemical or a nuclear plant (i.e., the liquid used to prevent the different parts of a plant from getting too hot) should have high specific heat.
- Q12.3(c): Explain why (c) Air pressure in a car tyre increases during driving.
- Q12.3(d): Explain why (d) The climate of a harbour town is more temperate than that of a town in a desert at the same latitude.
- Q12.4: A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume ?
- Q12.5: In changing the state of a gas adiabatically from an equilibrium state A to another equilibrium state B, an amount of work equal to 22.3 J is done on the system. If the gas is taken from state A to B via a process in which the net heat absorbed by the system is 9.35 cal, how much is the net work done by the system in the latter case ? (Take 1 cal = 4.19 J)
- Q12.6(a): Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following : (a) What is the final pressure of the gas in A and B ?
- Q12.6(b): Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following : (b) What is the change in internal energy of the gas ?
- Q12.6(c): Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following : (c) What is the change in the temperature of the gas ?
- Q12.7: A steam engine delivers $5.4 \times 10^8 J$ of work per minute and services $3.6 \times 10^9 J$ of heat per minute from its boiler. What is the efficiency of the engine? How much heat is wasted per minute?
- Q12.8: An electric heater supplies heat to a system at a rate of 100W. If system performs work at a rate of 75 joules per second. At what rate is the internal energy increasing?
- Q12.9: A thermodynamic system is taken from an original state to an intermediate state by the linear process shown in Fig. (12.13). Its volume is then reduced to the original value from E to F by an isobaric process. Calculate the total work done by the gas from D to E to F.
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