For a special dinner, you are gonna make a soup composed basically by 1350 cm^3 of water, with the addition of 240 cm^3 of whisky (ethyl alcohol content 50% + 50% water) when it's starts to cooking . However, one of the relatives invited can not consume alcohol, but you know that the alcohol content can be evaporate during the heating process.Using energy/mass balance, calculate how much time it would take to lower the alcohol content of the soup to inconspicuous levels.
Given:
The process is an open system (pressure = 101325 Pascal)
Cooking Initial temperature : 313,15 K
Whisky Initial temperature : 298,15 K
Ethyl alcohol boiling temperature: 351,52K
Ethyl alcohol Specific Heat(Cp (J/mol*K))=61,3 + (15,6*10^-2) ^T – (8,75*10^-5) * T^2 + (19,8*10^-9) * T^3
Water Specific Heat (Cp) = 4.186 joule/gram
External data like ethyl alcohol enthalpy of vaporization / enthalpy of saturated liquid can be used if necessary
To calculate the time it would take to lower the alcohol content of the soup to inconspicuous levels, we need to determine the rate at which the alcohol evaporates from the soup during the cooking process.
To calculate the rate of evaporation, we can use the concept of energy/mass balance. The energy transfer during the evaporation process can be expressed as:
Q = m * (ΔHv + Cp * ΔT)
Where:
Q = Energy transfer (in Joules)
m = Mass of the substance being evaporated (in grams)
ΔHv = Enthalpy of vaporization (in Joules/gram)
Cp = Specific heat capacity (in Joule/gram*K)
ΔT = Change in temperature (in Kelvin)
For this problem, we need to calculate the mass flow rate of alcohol vaporized from the soup. Let's assume that x grams of alcohol vaporize per second.
The initial mass of alcohol in the soup can be calculated using the volume and concentration of alcohol as follows:
m_initial_alcohol = (240 cm^3) * (50 g/cm^3) = 12000 grams
The initial mass of water in the soup can be calculated using the volume of water as follows:
m_initial_water = 1350 cm^3 * (1 g/cm^3) = 1350 grams
The total initial mass of the soup:
m_initial_total = m_initial_alcohol + m_initial_water
Now, let's calculate the rate of alcohol evaporation using the energy/mass balance equation:
Q_alcohol = (x g/s) * (ΔHv_alcohol + Cp_alcohol * ΔT)
For water, since the temperature remains constant during the heating process, there will be no change in temperature and thus no evaporation.
We know the enthalpy of vaporization for alcohol, but we also need the specific heat capacity of alcohol at the given temperature range. We can use the provided formula to calculate the specific heat capacity of alcohol (Cp_alcohol).
Now, we can set up an energy/mass balance equation for the system. The rate of alcohol evaporation should equal the rate at which alcohol is being removed from the system:
(x g/s) * (ΔHv_alcohol + Cp_alcohol * ΔT) = (x g/s) * (ΔHv_alcohol + Cp_water * ΔT)
Since the temperature change is the same for both alcohol and water, the term involving ΔT cancels out:
(x g/s) * ΔHv_alcohol = (x g/s) * ΔHv_water
Now, we can solve for x:
x = 0
This means that no alcohol will evaporate during the cooking process if the temperature remains constant. However, in reality, the alcohol will still evaporate to some extent due to the boiling point difference between alcohol and water.
To calculate the time it would take to lower the alcohol content to inconspicuous levels, we would need to consider additional factors such as the efficiency of heat transfer and the rate of evaporation at different temperatures. This calculation would require further data on heat transfer coefficients and knowledge of the system setup.
In conclusion, the exact time to lower the alcohol content of the soup cannot be determined based on the information provided.
To calculate the time it would take to lower the alcohol content of the soup to inconspicuous levels, we need to determine the rate at which alcohol evaporates and the amount of time required for the alcohol content to reach a desired level.
Step 1: Calculate the initial moles of alcohol in the whisky:
Moles of alcohol = Volume of whisky (cm^3) * Ethyl alcohol concentration
Moles of alcohol = 240 cm^3 * (50% / 100) = 120 moles
Step 2: Calculate the initial moles of water in the whisky:
Moles of water = Volume of whisky (cm^3) * Water concentration
Moles of water = 240 cm^3 * (50% / 100) = 120 moles
Step 3: Calculate the total moles of the mixture:
Total moles = Moles of alcohol + Moles of water
Total moles = 120 moles + 120 moles = 240 moles
Step 4: Calculate the initial mass of the mixture:
Initial mass = Total moles * Molar mass of the mixture
Assuming the molar mass of alcohol is approximately 46 g/mol and water is approximately 18 g/mol:
Initial mass = 240 moles * ((46 g/mol + 18 g/mol) / 2)
Initial mass = 240 moles * (64 g/mol)
Initial mass = 15360 g
Step 5: Determine the amount of alcohol that needs to evaporate to reach inconspicuous levels.
Let's assume that a 0.5% alcohol concentration is considered inconspicuous. Therefore, the desired moles of alcohol are given by:
Desired moles of alcohol = (0.5% / 100) * Total moles
Desired moles of alcohol = (0.5% / 100) * 240 moles
Desired moles of alcohol = 1.2 moles
Step 6: Calculate the amount of time required for alcohol content to reach inconspicuous levels:
We need to determine the rate of evaporation of alcohol. To do this, we'll use the energy/mass balance equation:
Rate of evaporation of alcohol = (Enthalpy of vaporization * Rate of mass transfer) / (Specific heat of alcohol + Specific heat of water)
Step 7: Calculate the enthalpy of vaporization of alcohol:
Given that the boiling temperature of ethyl alcohol is 351.52 K, we can assume that the enthalpy of vaporization is approximately 40.65 kJ/mol.
Step 8: Calculate the specific heat of alcohol at the initial temperature (313.15 K):
Cp (J/mol*K) = 61.3 + (15.6*10^-2 * T) - (8.75*10^-5 * T^2) + (19.8*10^-9 * T^3)
Cp (J/mol*K) = 61.3 + (15.6*10^-2 * 313.15) - (8.75*10^-5 * (313.15)^2) + (19.8*10^-9 * (313.15)^3)
Cp (J/mol*K) = 61.3 + 4.881 - 8.75*10^-5 * 97902.0225 + 19.8*10^-9 * 30664290.3897
Cp (J/mol*K) ≈ 61.3 + 4.881 - 8.546 + 6.071 ≈ 63.706 J/mol*K
Step 9: Calculate the rate of mass transfer:
Rate of mass transfer = (Moles of alcohol - Desired moles of alcohol) / Time
Step 10: Calculate the time required for alcohol content to reach inconspicuous levels:
Time = (Moles of alcohol - Desired moles of alcohol) / Rate of mass transfer
Step 11: Convert the result to the appropriate units if necessary.
By following these steps, you will be able to calculate the time it would take to lower the alcohol content of the soup to inconspicuous levels.