The pressure P1 and density d1 of diatomic gas (γ = 75) changes suddenly to P2(> P1) and d2 respectively during an adiabatic process. The temperature of the gas increases and becomes _______times of its initial temperature. (given d2d1 = 32)
Question
The pressure P1 and density d1 of diatomic gas (γ = 75) changes suddenly to P2(> P1) and d2 respectively during an adiabatic process. The temperature of the gas increases and becomes _______times of its initial temperature. (given d2d1 = 32)
Solution
The problem involves an adiabatic process, which is a process that occurs without transfer of heat or matter between a system and its surroundings. In an adiabatic process, the system either does work on its surroundings or has work done on it by its surroundings.
The equation for an adiabatic process for a diatomic gas is given by:
P1 * (d1)^γ = P2 * (d2)^γ
Where: P1 and P2 are the initial and final pressures, d1 and d2 are the initial and final densities, and γ is the heat capacity ratio (Cp/Cv), which is given as 7/5 for a diatomic gas.
We are given that P2 > P1, d2/d1 = 3/2, and we need to find the final temperature in terms of the initial temperature.
The equation for temperature and density for an ideal gas is given by:
T1/T2 = (d1/d2)^(γ-1)
Substituting the given values:
T1/T2 = (3/2)^(7/5 - 1) = (3/2)^(2/5) = 1.1487
Therefore, the final temperature is approximately 1.1487 times the initial temperature.
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