10 moles of a gas initially at temperature 300 K is compressed adiabatically from a volume of 2 cm3 to a volume of 325 cm3. To the nearest tenth of a joule what is the work done by the piston? (It is a monatomic ideal gas.)
To calculate the work done by the gas during adiabatic compression, we can use the equation:
W = -(n * Cv * ΔT)
Where:
W : Work done by the gas
n : Number of moles of the gas
Cv : Molar specific heat capacity at constant volume of the gas
ΔT : Change in temperature of the gas
Given values:
n = 10 moles
Cv = 3/2 R (for a monatomic ideal gas)
ΔT =T2 - T1
Since the initial and final temperatures are not given, we need to find them first.
To find the final temperature (T2), we can use the adiabatic equation:
P1 * V1^(γ) = P2 * V2^(γ)
Where:
P1 : Initial pressure of the gas
P2 : Final pressure of the gas
V1 : Initial volume of the gas
V2 : Final volume of the gas
γ : Adiabatic index (ratio of specific heats)
γ = Cp/Cv
Given values:
V1 = 2 cm^3
V2 = 325 cm^3
γ = Cp/Cv = 5/3 (for a monatomic ideal gas)
To find the initial pressure (P1), we can use the ideal gas law:
P1 * V1 = n * R * T1
Where:
R : Universal gas constant
T1 : Initial temperature of the gas
Given values:
T1 = 300 K
Now, we need to rearrange the adiabatic equation to solve for the final pressure (P2):
P2 = P1 * (V1/V2)^(γ)
Substituting the given values, we can calculate P2.
Next, we can calculate the final temperature using the ideal gas law:
T2 = P2 * V2 / (n * R)
Finally, we can substitute the calculated values of T2 and T1 into the equation for work done to find W.