suppose 125g of ethanol at 25C is mixed with 255g of ethanol at 84C at constant atmospheric pressure in a thermally insulated vessel. What is the Ssys for the process. The specific heat capacity for ethanol is 2.42 J/g K. Answer in units of J/K
To find the Ssys (change in entropy) for the process, we can use the equation:
ΔSsys = ∑(nRln(Vf/Vi)) + ∑(nCpln(Tf/Ti))
where:
- ΔSsys is the change in entropy of the system
- n is the number of moles
- R is the gas constant (8.314 J/(mol·K))
- ln is the natural logarithm
- Vf/Vi is the ratio of final volume to initial volume
- Tf/Ti is the ratio of final temperature to initial temperature
- Cp is the molar heat capacity at constant pressure
In this case, we are given the mass of ethanol, so we need to convert it to moles:
Molar mass of ethanol (C2H5OH) = 12.01 g/mol (for C) + 2(1.01 g/mol) (for H) + 16.00 g/mol (for O) + 1.01 g/mol (for H) = 46.08 g/mol
Number of moles of 125 g of ethanol = 125 g / 46.08 g/mol = 2.71 mol
Number of moles of 255 g of ethanol = 255 g / 46.08 g/mol = 5.53 mol
Now, let's calculate each term:
Term 1: ∑(nRln(Vf/Vi))
Since the vessel is thermally insulated, we assume that volume remains constant. Therefore, Vf/Vi = 1. So, this term would be zero.
Term 2: ∑(nCpln(Tf/Ti))
For this term, we need to calculate ΔT (change in temperature):
ΔT = Tf - Ti = 84°C - 25°C = 59°C
Now, we can calculate the second term:
∑(nCpln(Tf/Ti)) = (2.71 mol + 5.53 mol) × (2.42 J/g·K) × ln((84°C + 273.15 K)/(25°C + 273.15 K))
Convert the temperatures to Kelvin: 84°C + 273.15 = 357.15 K and 25°C + 273.15 = 298.15 K
∑(nCpln(Tf/Ti)) = 8.24 mol × (2.42 J/g·K) × ln(357.15 K / 298.15 K) = 8.24 mol × (2.42 J/g·K) × ln(1.197) ≈ 28.27 J/K
Now, summing the terms,we have:
ΔSsys = ∑(nRln(Vf/Vi)) + ∑(nCpln(Tf/Ti))
ΔSsys = Term 1 + Term 2
ΔSsys ≈ 0 + 28.27 J/K
Therefore, the ΔSsys for the process is approximately 28.27 J/K.