For Carbon dioxide, for a volume of 500 mL and temperature of 100 degrees celcius, calculate the pressures using the ideal gas law and the van der Waals equation. Explain any dicrepancies in the calculated values.
Ideal Gas Equation:
Van Der Waals Equation Pressure:
Explanation:
To calculate the pressure of Carbon dioxide (CO2) using the ideal gas law, we can use the formula:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant (0.0821 L · atm/(mol · K))
T = temperature in Kelvin (273 + temperature in Celsius)
Given:
Volume (V) = 500 mL = 0.5 L
Temperature (T) = 100 degrees Celsius = 373 K
However, we need the number of moles of CO2 in order to solve for the pressure. To find the number of moles, we need the molar mass of CO2. The molar mass of carbon dioxide is 44 g/mol.
We can calculate the number of moles (n) using the formula:
n = mass / molar mass
Now, let's suppose we have a certain mass of CO2, say 22 grams.
n = 22 g / 44 g/mol
n = 0.5 mol
Now, let's substitute the values into the ideal gas law equation:
P * 0.5 L = 0.5 mol * 0.0821 L · atm/(mol · K) * 373 K
Solving for P:
P = (0.5 mol * 0.0821 L · atm/(mol · K) * 373 K) / 0.5 L
P = 15.23 atm
So, according to the ideal gas law, the pressure of CO2 would be approximately 15.23 atmospheres.
Now let's calculate the pressure of CO2 using the van der Waals equation:
(P + a * (n/V)^2) * (V - n * b) = n * R * T
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant (0.0821 L · atm/(mol · K))
T = temperature in Kelvin (273 + temperature in Celsius)
a and b are the van der Waals constants (specific to each gas)
For CO2, the van der Waals constants are:
a = 3.59 atm L^2/mol^2
b = 0.0427 L/mol
Let's substitute the values into the equation:
(P + 3.59 atm L^2/mol^2 * (0.5 mol / 0.5 L)^2) * (0.5 L - 0.5 mol * 0.0427 L/mol) = 0.5 mol * 0.0821 L · atm/(mol · K) * 373 K
Simplifying the equation:
(P + 3.59 atm * (1/L)) * (0.5 L - 0.0427 L) = 0.5 mol * 0.0821 L · atm/(mol · K) * 373 K
Now, solve for P:
(P + 3.59 atm * (1/0.5)) * (0.4573 L) = 15.23 atm * 0.5 L
Simplifying further:
P + 7.18 atm * 0.4573 L = 7.615 atm * 0.5 L
P + 3.28 atm = 3.8085 atm
Thus:
P = 3.8085 atm - 3.28 atm
P = 0.53 atm
According to the van der Waals equation, the pressure of CO2 at those conditions would be approximately 0.53 atmospheres.
The discrepancy between the two calculated values can be attributed to the assumptions made in the ideal gas law. The ideal gas law assumes that gas molecules occupy negligible volume and do not interact with each other. In reality, gas molecules do have some volume and interact with each other, especially at high pressures and low temperatures.
The van der Waals equation takes into account these factors, specifically the volume of the gas molecules (represented by the parameter 'b') and the attractive forces between gas molecules (represented by the parameter 'a'). By considering these factors, the van der Waals equation provides a better approximation for real gases.
Therefore, a discrepancy between the two calculated values is expected, with the value obtained from the van der Waals equation being closer to the actual behavior of CO2 at high pressures and low temperatures.