an ideal monatomic gas is allow to expand slowly until its pressure is reduce to exactly half its original volume. By what factor those the volume change if the process is (1)isothermal (2) adiabatic
You must have meant "..until its pressure is reduce to exactly half its original VALUE". Pressure cannot equal volume. They have different dimensions.
(1) P*V = constant
V2/V1 = P1/P2 = 2
(2) P*V^5/3 = constant (for a monatomic gas)
(V2/V1)^(5/3) = P1/P2 = 2
V2/V1 = 2^(3/5) = 1.5157
To find out how the volume changes in each case, we can use the ideal gas law equation:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
(1) Isothermal process:
In an isothermal process, the temperature remains constant. Since the gas is ideal, according to the ideal gas law, PV = constant.
Initially, let's denote the original volume as V1 and the pressure as P1. When the pressure is reduced to half, the new pressure is P2 = P1/2.
Using the ideal gas law, we have:
P1V1 = P2V2
Since the temperature is constant, the equation becomes:
V2 = (P1/P2) * V1
V2 = (P1/(P1/2)) * V1
V2 = 2V1
Therefore, in an isothermal process, the volume doubles (changes by a factor of 2).
(2) Adiabatic process:
In an adiabatic process, there is no heat exchange with the surroundings. This means that the equation PV^γ = constant applies, where γ is the heat capacity ratio (γ = Cp/Cv).
For a monatomic ideal gas, γ = 5/3.
Using the same notations as before, we have:
P1V1^γ = P2V2^γ
Since the equation is in terms of pressure and volume, we need to eliminate pressure using the relation P2 = P1/2:
V1^γ = (P1/2)(V2/V1)^γ
(V2/V1)^γ = 2
Taking the γ-th root of both sides:
V2/V1 = 2^(1/γ)
V2/V1 = 2^(1/(5/3))
V2/V1 = 2^(3/5)
Therefore, in an adiabatic process, the volume changes by a factor of 2^(3/5).
To determine the volume change of an ideal monatomic gas during an isothermal or adiabatic process, we can refer to the ideal gas law and the corresponding equations for each process.
1. Isothermal process:
During an isothermal process, the temperature of the gas remains constant. The ideal gas law states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
Since the temperature is constant, we can rewrite the equation as:
P1V1 = P2V2
Where:
P1 = initial pressure
V1 = initial volume
P2 = final pressure
V2 = final volume
In this case, the final pressure is reduced to half its original value, so P2 = P1/2. Plugging this into the equation, we have:
P1V1 = (P1/2)V2
Simplifying the equation, we get:
V2 = 2V1
Therefore, the volume changes by a factor of 2 in the isothermal process.
2. Adiabatic process:
During an adiabatic process, no heat is exchanged between the gas and its surroundings. The equation for an adiabatic process is as follows:
PV^γ = constant
Where γ represents the heat capacity ratio, which is approximately 5/3 for a monatomic ideal gas.
Using the same initial and final pressure notation as before, we have:
P1V1^γ = P2V2^γ
Since the final pressure is reduced to half its original value, P2 = P1/2, and plugging it into the equation, we get:
P1V1^γ = (P1/2)V2^γ
Simplifying, we have:
V2^γ = 2^γ * V1^γ
Taking the γth root of both sides, we obtain:
V2 = (2^γ)^(1/γ) * V1
Since γ ≈ 5/3, (2^γ)^(1/γ) is approximately 1.587.
Therefore, the volume changes by a factor of approximately 1.587 in the adiabatic process.