A sample of natural gas is 85.2% methane, CH4, and 14.8% ethane, C2H6, by mass. What is the density of this gas mixture at 18.0°C and 771 mmHg? Assume ideal gas behavior.
To find the density of the gas mixture, we can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
First, let's calculate the number of moles of methane (CH4) and ethane (C2H6) present in the sample of natural gas.
Given:
Percent composition of methane by mass = 85.2%
Percent composition of ethane by mass = 14.8%
Total mass of the sample = 100 grams
Step 1: Calculate the mass of methane:
Mass of methane = 85.2% of 100 grams = 0.852 * 100 grams = 85.2 grams
Step 2: Calculate the mass of ethane:
Mass of ethane = 14.8% of 100 grams = 0.148 * 100 grams = 14.8 grams
Step 3: Convert the mass of methane and ethane to moles:
To do this, we need to know the molar mass of methane and ethane.
Molar mass of methane (CH4):
C: 12.01 g/mol
H: 1.01 g/mol
Total molar mass = 12.01 g/mol + (4 * 1.01 g/mol) = 16.05 g/mol
Number of moles of methane = Mass / Molar mass = 85.2 grams / 16.05 g/mol
Molar mass of ethane (C2H6):
C: 12.01 g/mol
H: 1.01 g/mol
Total molar mass = (2 * 12.01 g/mol) + (6 * 1.01 g/mol) = 30.07 g/mol
Number of moles of ethane = Mass / Molar mass = 14.8 grams / 30.07 g/mol
Step 4: Calculate the total moles of gas:
Total moles of gas = Moles of methane + Moles of ethane
Step 5: Use the ideal gas law to calculate the volume of the gas:
PV = nRT
V = (nRT) / P
Since we are asked to find the density, we need the volume to be in liters. Therefore, we need to convert the pressure to atmospheres (atm) and the temperature to Kelvin (K) as well.
Given:
Pressure (P) = 771 mmHg
Temperature (T) = 18.0°C = 18.0 + 273.15 = 291.15 K
R = 0.0821 L·atm/(mol·K) (ideal gas constant)
Step 6: Convert the pressure and temperature:
Pressure (P) = 771 mmHg / 760 mmHg/atm (since 1 atm = 760 mmHg)
Temperature (T) = 291.15 K
Step 7: Substitute the values into the equation and solve for V:
V = (nRT) / P
Step 8: Calculate the density of the gas:
Density = Mass / Volume = Total mass of the sample / Volume
Now we can calculate the density of the gas mixture.
To calculate the density of a gas mixture, we need to use the ideal gas law equation:
PV = nRT
Where:
P is the pressure of the gas in atm
V is the volume of the gas in liters
n is the number of moles of the gas
R is the ideal gas constant (0.0821 L·atm/(mol·K))
T is the temperature of the gas in Kelvin
First, we need to convert the given temperature from Celsius to Kelvin. To do this, we add 273.15 to the Celsius temperature:
T = 18.0°C + 273.15 = 291.15 K
Next, we need to convert the pressure from mmHg to atm. Since 1 atm is equal to 760 mmHg, we divide the given pressure by 760:
P = 771 mmHg / 760 = 1.0145 atm
Now, we need to calculate the number of moles of methane and ethane in the mixture. We'll assume we have 100 grams of the mixture to make the calculations easier.
The molar mass of methane (CH4) is 16.04 g/mol, and the molar mass of ethane (C2H6) is 30.07 g/mol.
The mass of methane in the mixture is: 85.2 g * (0.852) = 72.34 g
The mass of ethane in the mixture is: 85.2 g * (0.148) = 12.62 g
Now we can calculate the number of moles using the molar masses:
Number of moles of methane = 72.34 g / 16.04 g/mol = 4.51 mol
Number of moles of ethane = 12.62 g / 30.07 g/mol = 0.42 mol
To find the total number of moles, we sum the moles of methane and ethane:
Total number of moles = 4.51 mol + 0.42 mol = 4.93 mol
Now we can calculate the volume using the ideal gas law. Since we have 4.93 moles of gas, we need to find the volume that this gas occupies at the given temperature and pressure:
PV = nRT
V = (nRT) / P
V = (4.93 mol) * (0.0821 L·atm/(mol·K)) * (291.15 K) / (1.0145 atm)
Calculating this expression gives us the volume in liters.
Finally, since we assumed we had 100 grams of the gas mixture, we can calculate the density:
Density = mass / volume
Density = (100 g) / (volume in liters)
Substituting the calculated volume, we get the final density in g/L.
Please note that the above calculations assume the gas behaves ideally, which is an approximation. In reality, there might be minor deviations from ideal behavior.
I would calculate an "average" molar mass.
Take a 1 g sample you will have 0.852g CH4 and 0.148 g C2H6.
(0.852 x molar mass CH4) + (0.148 x molar mass C2H6) = molar mass of the mixture.
Then P*molar mass = dRT
d = density in g/L.