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Initial pressure and volume of a gas are P and V respectively. First it is expanded isothermally to volume 4V and then compressed adiabatically to volume V. The final pressure of gas will be (γ =1.5)
- a)P
- b)2P
- c)4P
- d)8P
Correct answer is option 'B'. Can you explain this answer?
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Initial pressure and volume of a gas are P and V respectively. First i...
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Initial pressure and volume of a gas are P and V respectively. First i...
Given:
Initial pressure = P
Initial volume = V
Isothermal expansion to volume 4V
Adiabatic compression to volume V
Ratio of specific heats = 1.5
To find:
Final pressure of the gas
Solution:
Isothermal expansion:
During the isothermal expansion, the temperature of the gas remains constant. Therefore, we can use the ideal gas equation to relate the pressure and volume of the gas.
PV = nRT
where n is the number of moles of the gas and R is the gas constant.
Since the temperature remains constant, we can write:
P1V1 = P2V2
where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
Substituting the given values, we get:
P(V) = (P/4)(4V) = PV
Therefore, the pressure of the gas remains constant during the isothermal expansion and is equal to the initial pressure P.
Adiabatic compression:
During the adiabatic compression, no heat is exchanged between the gas and its surroundings. Therefore, the process can be described by the following relation between pressure and volume:
P(V)^(γ) = constant
where γ is the ratio of specific heats.
Substituting the initial and final volumes, we get:
P(V)^(γ) = P(4V)^(γ) = P(V)^(γ) / (4^(γ-1))
Simplifying, we get:
P(V)^(γ) = (1/4)P(V)^(γ) or P = 4P/4^(γ-1)
Substituting γ = 1.5, we get:
P = 4P/2.25 = 8P/3
Therefore, the final pressure of the gas is 2 times the initial pressure or 2P.
Hence, the correct option is (B).
Initial pressure = P
Initial volume = V
Isothermal expansion to volume 4V
Adiabatic compression to volume V
Ratio of specific heats = 1.5
To find:
Final pressure of the gas
Solution:
Isothermal expansion:
During the isothermal expansion, the temperature of the gas remains constant. Therefore, we can use the ideal gas equation to relate the pressure and volume of the gas.
PV = nRT
where n is the number of moles of the gas and R is the gas constant.
Since the temperature remains constant, we can write:
P1V1 = P2V2
where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
Substituting the given values, we get:
P(V) = (P/4)(4V) = PV
Therefore, the pressure of the gas remains constant during the isothermal expansion and is equal to the initial pressure P.
Adiabatic compression:
During the adiabatic compression, no heat is exchanged between the gas and its surroundings. Therefore, the process can be described by the following relation between pressure and volume:
P(V)^(γ) = constant
where γ is the ratio of specific heats.
Substituting the initial and final volumes, we get:
P(V)^(γ) = P(4V)^(γ) = P(V)^(γ) / (4^(γ-1))
Simplifying, we get:
P(V)^(γ) = (1/4)P(V)^(γ) or P = 4P/4^(γ-1)
Substituting γ = 1.5, we get:
P = 4P/2.25 = 8P/3
Therefore, the final pressure of the gas is 2 times the initial pressure or 2P.
Hence, the correct option is (B).
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