an ideal monatomic gas is allow to expand slowly until its presssure is reduced to exactly half its original value. By what factor does the volume change if the pressure is (1) isothermal (2) adiabatic
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To find the factor by which the volume changes for an ideal monatomic gas when the pressure is reduced to exactly half its original value, we can use the gas laws – specifically, the equations for isothermal and adiabatic processes.
1. Isothermal Process:
In an isothermal process, the temperature of the gas remains constant. According to Boyle's Law, for a given amount of gas at a constant temperature, the product of its pressure (P) and volume (V) is constant. Mathematically, P1V1 = P2V2, where P1 is the initial pressure, V1 is the initial volume, P2 is the final pressure, and V2 is the final volume.
In this case, the final pressure (P2) is half the initial pressure (P1). So, the equation becomes:
P1V1 = (P1/2) V2
To find the factor by which the volume changes, we need to solve for V2 in terms of V1:
V2 = 2V1
Therefore, the volume (V2) will be twice the initial volume (V1).
2. Adiabatic Process:
In an adiabatic process, no heat is exchanged between the gas and its surroundings. For an ideal gas following an adiabatic process, the relationship between pressure and volume is given by the adiabatic equation:
P1V1^γ = P2V2^γ
Here, γ represents the heat capacity ratio, which is approximately 5/3 for a monatomic gas.
In this case, the final pressure (P2) is half the initial pressure (P1). So, the equation becomes:
P1V1^γ = (P1/2) V2^γ
To find the factor by which the volume changes, we need to solve for V2 in terms of V1:
V2^γ = (2)^(γ-1) V1^γ
Taking the γth root of both sides, we get:
V2 = (2)^(1-γ/γ) V1
For a monatomic gas, γ = 5/3. Substituting this value, we get:
V2 = (2)^(3/5) V1
Therefore, the volume (V2) will change by a factor of the fifth root of 2, approximately 1.1487, compared to the initial volume (V1).
So, in summary:
1. For an isothermal process, the volume will change by a factor of 2.
2. For an adiabatic process, the volume will change by a factor of approximately 1.1487 (fifth root of 2).
To find the factor by which the volume changes for an ideal monatomic gas undergoing different processes, we need to use the ideal gas law and the specific heat ratio for the gas.
Let's denote the initial pressure as P1 and the final pressure as P2, and the initial volume as V1 and the final volume as V2.
(1) Isothermal Process:
In an isothermal process, the temperature remains constant. The ideal gas law for an isothermal process is given by:
P1 * V1 = P2 * V2
Since the pressure is reduced to exactly half its original value, we can rewrite this equation as:
P1 * V1 = (1/2) * P1 * V2
By dividing both sides by P1, we get:
V1 = (1/2) * V2
Therefore, the volume decreases by a factor of 1/2.
(2) Adiabatic Process:
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure and volume for an adiabatic process is governed by the following equation:
P1 * V1^gamma = P2 * V2^gamma
where gamma is the specific heat ratio, which is equal to 5/3 for monatomic gases.
Since the pressure is reduced to exactly half its original value, we can rewrite this equation as:
P1 * V1^gamma = (1/2) * P1 * V2^gamma
By dividing both sides by P1 and taking the gamma root, we get:
V1^(gamma-1) = (1/2) * V2^(gamma-1)
For a monatomic gas, gamma-1 is equal to 2/3, so we have:
V1^(2/3) = (1/2) * V2^(2/3)
By raising both sides to the power of (3/2), we get:
V1 = (1/2)^(3/2) * V2
Simplifying further, we have:
V1 = (1/2) * sqrt(1/2) * V2
V1 = (1/2) * (1/sqrt(2)) * V2
Therefore, the volume decreases by a factor of (1/2) * (1/sqrt(2)).
To summarize, the volume changes by the following factors:
(1) Isothermal process: 1/2
(2) Adiabatic process: (1/2) * (1/sqrt(2))