1. From data below, calculate the total heat (in J) associated with the conversion of 0.514 mol ethanol gas (C2H6O) at 451°C and 1 atm to liquid ethanol at 25.0°C and 1 atm. (Pay attention to the sign of the heat.)
Boiling point at 1 atm 78.5°C
cgas 1.43 J/g°C
cliquid 2.45 J/g°C
H°vap 40.5 kJ/mol
2. A 0.75 L bottle is cleaned, dried, and closed in a room where the air is 20°C and 42% relative humidity (that is, the water vapor in the air is 0.42 of the equilibrium vapor pressure at 20°C). The bottle is brought outside and stored at 0.0°C.
What mass of water condenses inside the bottle?
To calculate the total heat associated with the conversion of ethanol gas to liquid ethanol, we need to consider the heat absorbed or released during various steps.
Step 1: Cooling the ethanol gas from 451°C to its boiling point at 78.5°C
In this step, heat is released. We can calculate the heat using the formula:
q1 = n × cgas × ΔT
Here,
n = number of moles of ethanol gas = 0.514 mol
cgas = specific heat capacity of ethanol gas = 1.43 J/g°C
ΔT = change in temperature = (78.5°C - 451°C) = -372.5°C
Convert the number of moles to grams:
molar mass of ethanol (C2H6O) = 46.07 g/mol
mass of ethanol gas = n × molar mass = 0.514 mol × 46.07 g/mol = 23.68 g
Substitute the values into the formula:
q1 = 23.68 g × 1.43 J/g°C × (-372.5°C) = -12269.96 J
Step 2: Condensation of ethanol gas to liquid ethanol
In this step, heat is released during the formation of liquid ethanol. The heat released is the enthalpy of vaporization (H°vap) multiplied by the number of moles.
We are given that H°vap = 40.5 kJ/mol.
Convert H°vap to joules:
1 kJ = 1000 J
H°vap = 40.5 kJ/mol × 1000 J/kJ = 40500 J/mol
Multiply H°vap by the number of moles:
q2 = n × H°vap
Substitute the values into the formula:
q2 = 0.514 mol × 40500 J/mol = 20847 J
Step 3: Cooling the liquid ethanol from its boiling point to 25.0°C
In this step, heat is released. We can calculate the heat using the formula:
q3 = m × cliquid × ΔT
Here,
m = mass of liquid ethanol = 23.68 g (which was calculated in Step 1)
cliquid = specific heat capacity of liquid ethanol = 2.45 J/g°C
ΔT = change in temperature = (25.0°C - 78.5°C) = -53.5°C
Substitute the values into the formula:
q3 = 23.68 g × 2.45 J/g°C × (-53.5°C) = -3096.18 J
Total Heat: The total heat is the sum of the individual heats from each step.
Total Heat = q1 + q2 + q3
Total Heat = -12269.96 J + 20847 J - 3096.18 J = 5281.86 J
Therefore, the total heat associated with the conversion of 0.514 mol of ethanol gas at 451°C and 1 atm to liquid ethanol at 25.0°C and 1 atm is 5281.86 J (with the negative sign indicating the release of heat during the process).
Now, let's move on to the second question about the condensation of water inside a bottle.
To calculate the mass of water that condenses inside the bottle, we need to consider the change in temperature and the relative humidity.
Step 1: Using the relative humidity to calculate the partial pressure of water vapor
Given relative humidity = 42%
Equilibrium vapor pressure at 20°C = P₀
Partial pressure of water vapor in the air = 0.42 × P₀
Step 2: Converting the partial pressure to moles of water vapor
Using the ideal gas law, PV = nRT, where P = partial pressure of gas, V = volume, n = number of moles, R = ideal gas constant, and T = temperature in Kelvin.
We can rearrange the equation to solve for n:
n = PV / RT = (0.42 × P₀) × (0.75 L) / (0.0821 L·atm/mol·K × (20°C + 273.15) K)
Step 3: Converting moles of water vapor to mass of water
Using the molar mass of water (H₂O) = 18.02 g/mol:
mass of water = n × molar mass = (0.42 × P₀) × (0.75 L) / (0.0821 L·atm/mol·K × (20°C + 273.15) K) × 18.02 g/mol
Substitute the values into the formula to calculate the mass of water condensing inside the bottle.