Answer: (a) The final volume of the gas is 0.078 m^3.
(b) The work done by the gas is -2.2997 kJ.
(c) The thermal energy transferred is 2.2997 kJ.
Explanation: (a) The process is isothermal, which means the temperature remains constant during the compression. Therefore, 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 gas constant, and T is the temperature.
Since the process is isothermal, T is constant, and we can write:
P1V1 = P2V2
where subscripts 1 and 2 refer to the initial and final states, respectively.
We are given that n = 2 mol, P1 = 0.42 atm, P2 = 1.88 atm, and T = 257 K. Therefore, we can solve for V2:
V2 = V1 * P1/P2 = (nRT)/P2 * P1
Substituting the values, we get:
V2 = (2 mol * 8.31451 J/K·mol * 257 K) / (1.88 atm) * (0.42 atm) = 0.078 m^3
Therefore, the final volume of the gas is 0.078 m^3.
(b) The work done by the gas during an isothermal process is given by:
W = -nRT ln(P2/P1)
Substituting the values, we get:
W = -(2 mol) * (8.31451 J/K·mol) * (257 K) * ln(1.88/0.42) = -2299.7 J
Therefore, the work done by the gas is -2299.7 J or -2.2997 kJ (to three significant figures).
(c) Since the process is isothermal, the thermal energy transferred is equal to the work done by the gas:
Q = -W = 2.2997 kJ
Therefore, the thermal energy transferred is 2.2997 kJ.