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?
1. To calculate the total heat associated with the conversion of ethanol gas to liquid ethanol, we need to consider the different steps involved in the process.
Step 1: Calculate the heat required to cool down the ethanol gas from 451°C to its boiling point at 1 atm, which is 78.5°C. Since ethanol is in the gas phase, we can use the heat capacity of the gas (cgas) to calculate this heat.
Heat1 = m × cgas × ΔT1
where
m is the mass of ethanol gas (C2H6O) in grams
cgas is the specific heat capacity of the gas (1.43 J/g°C)
ΔT1 is the change in temperature (451°C - 78.5°C)
Step 2: Calculate the heat required to vaporize the liquid ethanol at its boiling point. This is known as the enthalpy of vaporization (H°vap) and is given in kJ/mol. We need to convert moles of ethanol to grams and then multiply by the enthalpy of vaporization.
Heat2 = n × H°vap
where
n is the number of moles of ethanol (C2H6O)
H°vap is the enthalpy of vaporization (40.5 kJ/mol)
Step 3: Calculate the heat released when the vaporized ethanol condenses to liquid ethanol at 25.0°C. This heat can be calculated using the heat capacity of the liquid (cliquid).
Heat3 = m × cliquid × ΔT2
where
m is the mass of condensed ethanol in grams
cliquid is the specific heat capacity of the liquid (2.45 J/g°C)
ΔT2 is the change in temperature (25.0°C - 78.5°C)
Finally, to find the total heat, we can sum up all the heats calculated in the above steps. The signs of the heat should also be taken into account to indicate whether heat is gained or lost.
Total Heat = Heat1 + Heat2 + Heat3
2. To calculate the mass of water that condenses inside the bottle when brought outside and stored at 0.0°C, we need to consider the change in temperature and relative humidity.
Step 1: Calculate the equilibrium vapor pressure of water at 20°C using the given relative humidity. The equilibrium vapor pressure of water at a given temperature can be found in water vapor pressure tables.
Step 2: Calculate the actual vapor pressure of water at 20°C using the equilibrium vapor pressure and relative humidity.
Actual Vapor Pressure = Equilibrium Vapor Pressure × Relative Humidity
Step 3: Calculate the mass of water that condenses when the temperature is lowered to 0.0°C. We can use the Clausius-Clapeyron equation to calculate the change in vapor pressure.
ln(P2/P1) = ΔHvap/R × (1/T1 - 1/T2)
where
P2 is the actual vapor pressure at 0.0°C
P1 is the actual vapor pressure at 20°C
ΔHvap is the enthalpy of vaporization of water
R is the gas constant
T1 is the initial temperature (20°C + 273.15 K)
T2 is the final temperature (0.0°C + 273.15 K)
Step 4: Calculate the mass of water using the ideal gas law.
Mass = (P2 × V) / (R × T2)
where
P2 is the actual vapor pressure at 0.0°C
V is the volume of the bottle in liters
R is the gas constant
T2 is the final temperature (0.0°C + 273.15 K)
By following these steps and using the given data, you can calculate the total heat associated with the conversion of ethanol gas to liquid ethanol and the mass of water that condenses inside the bottle.