An ideal gas undergoes an isothermal process at a temperature of 300 K. The gas starts at an initial pressure of 2.0 atm and expands to a final volume of 8.0 L. During the process, the gas absorbs 400 J of energy as heat. Assume the gas remains ideal throughout the process.
(a) Determine the final pressure of the gas.
(b) Calculate the work done by the gas during this process.
(c) Calculate the change in internal energy of the gas.
Assume all values are exact and use the following gas constant: R = 8.31 J/(mol·K).
Given:
(a) To determine the final pressure (P₂) of the gas, we can use the ideal gas law equation:
P₁ × V₁ = P₂ × V₂
Since the process is isothermal, the initial and final temperatures are the same. Therefore, we can rewrite the equation using the ideal gas law:
P₁ × V₁/T₁ = P₂ × V₂/T₂
Since we know P₁, V₁, T₁, and V₂, we can substitute these values into the equation:
2.0 atm × V₁/300 K = P₂ × 8.0 L/300 K
Simplifying the equation:
V₁/300 = P₂ × 8.0/300
V₁ = P₂ × 8.0/300
P₂ = P₁ × V₁/V₂
P₂ = 2.0 atm × 8.0 L/300 L
Calculating P₂:
P₂ = (2.0 × 8.0)/300
P₂ = 0.0533 atm
Therefore, the final pressure of the gas is approximately 0.0533 atm.
(b) To calculate the work done by the gas during this process, we can use the equation:
Work (W) = -q (since heat is absorbed)
Substituting the given values:
W = -400 J
Therefore, the work done by the gas during this process is -400 J.
(c) The change in internal energy (ΔU) of the gas can be calculated using the first law of thermodynamics:
ΔU = q + W
Substituting the given values:
ΔU = 400 J + (-400 J)
ΔU = 0 J
Therefore, the change in internal energy of the gas is 0 J.
In summary, (a) The final pressure of the gas is approximately 0.0533 atm. (b) The work done by the gas during this process is -400 J. (c) The change in internal energy of the gas is 0 J.