Final Answer:
The work done by the gas is 4.0 kJ.
Step-by-step explanation:
For an adiabatic process of an ideal gas, we can use the following equation to relate the initial and final pressures (P1 and P2) and volumes (V1 and V2):
P1V1γ = P2V2γ
where γ is the heat capacity ratio, which is 5/3 for a monatomic gas.
We are given that the initial volume is 10 L, the final volume is 40 L, and the initial temperature is 500 K. We can use the ideal gas law to relate the initial pressure to the initial volume and temperature:
PV = nRT
where P is the pressure (in atm), V is the volume (in L), n is the number of moles (in mol), R is the ideal gas constant (in L·atm/mol·K), and T is the temperature (in K).
Since the gas is monatomic, we know that n = 1 mol. We can also find the ideal gas constant from the following equation:
R = 0.0821 L·atm/mol·K
Plugging in the values we know, we can solve for the initial pressure:
P1 = (nRT)/V1 = (1 mol)(0.0821 L·atm/mol·K)(500 K)/10 L = 4.11 atm
Now we can plug in the values for P1, V1, and γ into the adiabatic equation to solve for P2:
P1V1γ = P2V2γ
(4.11 atm)(10 L)^(5/3) = P2(40 L)^(5/3)
Solving for P2, we get:
P2 = 1.03 atm
Now we can use the work formula for an adiabatic process:
W = -(P2V2 - P1V1)
Plugging in the values we know, we get:
W = -(1.03 atm)(40 L) - (4.11 atm)(10 L) = -4.0 kJ
Since work is a negative quantity when the system does work on the surroundings, we can take the absolute value of the result to get the work done by the gas:
|W| = 4.0 kJ
Therefore, the work done by the gas is 4.0 kJ.