A 3.50 liter ballooncontains air at 65% relative humidity at 22 degrees C at 1.0 atm pressure. A small peice of Na metal is introduced to the balloon to remove all of the water vapor. What would the new volume be for the dry air in the balloon?
To find the new volume of the dry air in the balloon, we'll need to understand the concept of ideal gas law and how it relates to humidity, temperature, and pressure.
The ideal gas law is expressed by the equation: PV = nRT, where:
P = pressure (in atm)
V = volume (in liters)
n = number of moles of gas
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature (in Kelvin)
Relative humidity (RH) is the ratio of the actual amount of water vapor present in the air to the maximum amount of water vapor that the air could hold at a given temperature. It is expressed as a percentage.
First, we need to calculate the amount of water vapor in the balloon. We know that the balloon has a capacity of 3.50 liters and contains air at 65% relative humidity at 22 degrees Celsius. To do this, we'll use the following steps:
Step 1: Convert temperature to Kelvin
Temperature in Kelvin (T) = Celsius temperature + 273.15
T = 22 + 273.15 = 295.15 K
Step 2: Calculate the maximum amount of water vapor at 22 degrees Celsius
To calculate the maximum amount of water vapor, we can use the Clausius-Clapeyron equation, which relates the vapor pressure of water to temperature:
ln(Pvap) = -C + (ΔHvap/R)(1/T)
where:
Pvap = vapor pressure of water at a given temperature
C = constant
ΔHvap = enthalpy of vaporization of water (40.7 kJ/mol)
R = ideal gas constant (8.314 J/(mol·K))
T = temperature in Kelvin
We'll use the equation to calculate the vapor pressure of water at 22 degrees Celsius (295.15 K):
ln(Pvap) = -C + (ΔHvap/R)(1/T)
Since we're interested in the relative humidity of 65%, the vapor pressure of water (Pvap) in the air is equal to 65% of the maximum vapor pressure at the given temperature.
Step 3: Calculate the amount of water vapor in moles
Using the ideal gas law, we can calculate the number of moles of water vapor in the balloon:
n = PV / (RT)
where:
P = vapor pressure of water (calculated in step 2)
V = volume of the balloon (3.50 L)
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature in Kelvin (295.15 K)
Now that we have calculated the moles of water vapor, we can determine the volume occupied by the dry air in the balloon by subtracting the volume occupied by the water vapor.
Step 4: Calculate the moles of dry air
To find the moles of dry air, we subtract the moles of water vapor from the total amount of moles of gas in the balloon. However, we need to consider that the Na metal introduced will react with the water vapor to produce hydrogen gas (H2) and sodium hydroxide (NaOH).
Na + H2O -> H2 + NaOH
Since the reaction consumes 1 mole of Na and 1 mole of H2O to produce 1 mole of H2, the decrease in moles of water vapor will be equal to the decrease in moles of Na. The moles of dry air will be the total moles of gas in the balloon minus the moles of Na.
Step 5: Calculate the new volume of the dry air
With the moles of dry air calculated in step 4, we can use the ideal gas law to calculate the new volume of the dry air:
V = (nRT) / P
where:
n = moles of dry air (calculated in step 4)
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature in Kelvin (295.15 K)
P = pressure (1.0 atm)
Substitute the calculated values into the equation to find the new volume of the dry air in the balloon.
By following these steps and performing the calculations, you will be able to determine the new volume of the dry air in the balloon after the removal of water vapor.