Five moles of a monatomic ideal gas expand adiabatically and its temperature decreases fa) the work done by the gas rom 395 K to 298 K. Calculatet
he change in its internal energy
To calculate the work done by the gas during an adiabatic expansion, we need to use the adiabatic equation:
W = (P2V2 - P1V1) / (γ - 1),
where W is the work done, P1 and P2 are the initial and final pressures, V1 and V2 are the initial and final volumes, and γ is the adiabatic index (also known as the heat capacity ratio).
Since the gas is monatomic, its adiabatic index is γ = 5/3.
To determine the initial and final pressures, we can use the ideal gas law:
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.
For the initial state:
P1 = (nRT1) / V1.
For the final state:
P2 = (nRT2) / V2.
Now, we can substitute these values into the adiabatic work equation:
W = [(nRT2 / V2)(V2) - (nRT1 / V1)(V1)] / (γ - 1).
Simplifying further:
W = nR(T2 - T1) / (γ - 1),
where R is the ideal gas constant (8.314 J/mol K).
Substituting the given values:
n = 5 moles,
T1 = 395 K,
T2 = 298 K,
γ = 5/3,
R = 8.314 J/mol K.
W = 5 * 8.314 * (298 - 395) / (5/3 - 1).
Calculating the value of W:
W = -8,301.9 J.
The negative sign indicates that work is done by the gas during its adiabatic expansion.
To calculate the change in internal energy (ΔU), we can use the First Law of Thermodynamics:
ΔU = Q - W,
where Q is the heat transferred.
Since the process is adiabatic (no heat is transferred), Q = 0.
Therefore, ΔU = -W.
Substituting the value of W:
ΔU = -(-8,301.9 J) = 8,301.9 J.
So, the change in internal energy for this adiabatic expansion is 8,301.9 J.