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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 ?
Solution :
Step 1: Analyzing the Thermodynamic Process
The problem states that the cylinder walls and the piston are made of heat insulators. This implies that no heat exchange occurs between the system and the surroundings ($dQ = 0$). [Per the definition of an adiabatic process]. Consequently, the compression of the hydrogen gas follows the adiabatic relation between pressure ($P$) and volume ($V$).
Step 2: Defining Physical Constants for Hydrogen
Hydrogen ($H_2$) is a diatomic gas. For a diatomic gas, the degree of freedom ($f$) is 5 at room temperature. The adiabatic exponent ($\gamma$), defined as the ratio of molar heat capacity at constant pressure ($C_p$) to constant volume ($C_v$), is calculated as follows:
$\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f} = 1 + \frac{2}{5} = 1.4$
Step 3: Applying the Adiabatic Gas Law
The state equation for an adiabatic process is given by: $P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$ [Per the Adiabatic Equation of State]
Given the condition that the gas is compressed to half its original volume: $V_2 = \frac{1}{2} V_1$
Step 4: Algebraic Derivation of the Pressure Factor
Rearranging the adiabatic equation to find the ratio of final pressure ($P_2$) to initial pressure ($P_1$): $\frac{P_2}{P_1} = \left( \frac{V_1}{V_2} \right)^{\gamma}$
Substituting $V_2 = \frac{V_1}{2}$ into the equation: $\frac{P_2}{P_1} = \left( \frac{V_1}{V_1 / 2} \right)^{1.4}$
$\frac{P_2}{P_1} = (2)^{1.4}$
Step 5: Numerical Calculation
To evaluate $(2)^{1.4}$:
$(2)^{1.4} \approx 2.639$
Summary Table of Parameters
| Parameter | Value |
|---|---|
| Adiabatic Exponent ($\gamma$) | $1.4$ |
| Volume Compression Ratio | $2$ |
| Pressure Factor | $2^{1.4} \approx 2.64$ |
Final Solution: The pressure of the gas increases by a factor of approximately 2.64.
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.
- Q12.5: 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 ?
- Q12.6(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 ?
- 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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