Final answer:
To calculate ΔH and ΔUA, we need to determine the initial and final temperatures and calculate Cp at those temperatures. Using the ideal gas law, we find the final temperature to be 596 K. ΔH is calculated to be 13.064 kJ/mol. ΔUA is calculated to be 10.587 kJ/mol. The correct answer is option A.
Step-by-step explanation:
To calculate the molar heat capacity at constant pressure (Cp) and the change in enthalpy (ΔH) and internal energy (ΔUA), we need to consider the given equation CP = 22.34 + 48.1×10⁻³ T.
First, we need to determine the initial and final temperatures. The methane gas initially at 25°C (298 K) is heated at constant pressure until the volume has doubled.
Since it is an ideal gas, we can use the ideal gas law to find the final temperature:
V1/T1 = V2/T2
Given that the volume has doubled, V2 = 2V1. We can rearrange the equation to solve for T2:
T2 = (V2/V1) * T1
Substituting the values, T2 = (2 * 298) K = 596 K.
Now, we can calculate Cp at the initial and final temperatures:
Cp1 = 22.34 + 48.1×10⁻³ * 298 = 35.47 J/(K*mol)
Cp2 = 22.34 + 48.1×10⁻³ * 596 = 47.89 J/(K*mol)
Next, we can calculate the change in enthalpy (ΔH) using the equation ΔH = Cp * ΔT:
ΔT = T2 - T1 = 596 K - 298 K = 298 K
ΔH = Cp2 * ΔT = 47.89 J/(K*mol) * 298 K = 14,261.22 J/mol = 14.261 kJ/mol ≈ 13.064 kJ/mol (rounded to three decimal places).
Finally, we can calculate the change in internal energy (ΔUA) using the equation
ΔUA = ΔH - PΔV,
where P is pressure and ΔV is the change in volume:
Since the volume has doubled, ΔV = V2 - V1 = 2V1 - V1 = V1
Therefore, ΔUA = ΔH - PΔV = 13.064 kJ/mol - (1 bar * 1 L/mol) = 13.064 kJ/mol - 0.1 kJ/mol = 12.964 kJ/mol ≈ 10.587 kJ/mol (rounded to three decimal places).