One mole of an ideal gas whose adiabatic exponent equals R undergoes a...
To solve this problem, let's start with the ideal gas law:
PV = nRT
where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
Since we're given that the gas pressure relates to the temperature as P = aT, we can substitute this into the ideal gas law equation:
(aT)V = nRT
Next, let's consider the adiabatic exponent, γ. For an ideal gas, γ is defined as the ratio of specific heat capacities:
γ = Cp/Cv
where:
Cp = specific heat capacity at constant pressure
Cv = specific heat capacity at constant volume
Since the gas is an ideal gas, γ is constant. Given that the adiabatic exponent equals R, we have:
γ = R
Now, let's consider an adiabatic process, where no heat is exchanged with the surroundings. In an adiabatic process, the relationship between pressure and volume is given by:
PV^γ = constant
Substituting the equation relating pressure to temperature, we have:
(aT)V^γ = constant
Since we are given that there is one mole of gas, we can substitute n = 1 into the ideal gas law equation:
(aT)V = RT
Rearranging this equation, we have:
V = RT/aT
Simplifying, we find:
V = R/a
So, in an adiabatic process, the volume of one mole of gas is directly proportional to the ideal gas constant divided by the constant of proportionality, a.
Note: The adiabatic exponent being equal to R is not a typical scenario for an ideal gas. Usually, γ takes on values between 1 and 2, depending on the gas.
One mole of an ideal gas whose adiabatic exponent equals R undergoes a...