A sample of a breathing mixture for divers comtained 34.3% helium, 51.7% nitrogen, and 14% oxygen, by mass. What is the density of this mixture at 22degree celcius and 755mmHG?
If you take a 100 g sample, you will have 34.3 g He, 51.7 g N2, and 14 g O2. Convert those to moles (moles = g/molar mass), then use PV = nRT and solve for V. (n = total moles of the three gases.)
Then density = mass/volume
To calculate the density of the breathing mixture, we can use the Ideal Gas Law equation:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
First, we need to find the molar mass of the breathing mixture. To do this, we can calculate the average molar mass using the percentages of the different gases present.
The molar mass of helium (He) is approximately 4 g/mol.
The molar mass of nitrogen (N2) is approximately 28 g/mol.
The molar mass of oxygen (O2) is approximately 32 g/mol.
The molar mass of the breathing mixture can be calculated using the following formula:
Molar mass = (Percent He / 100) × Molar mass He + (Percent N2 / 100) × Molar mass N2 + (Percent O2 / 100) × Molar mass O2
Molar mass = (0.343 × 4) + (0.517 × 28) + (0.140 × 32)
Molar mass = 1.372 + 14.476 + 4.48
Molar mass = 20.328 g/mol
Now, we can substitute the values into the Ideal Gas Law equation and solve for density.
PV = nRT
The molar volume (Vm) can be calculated as follows:
Vm = RT / P
Vm = (0.0821 L/mol·K) × (22 + 273.15) / (755 mmHg × 1 atm/760 mmHg)
Vm = (0.0821 L/mol·K) × 295.15 K / (0.9934 atm)
Vm = 0.0246153 L/mol
Now, we can calculate the density (D) using the formula:
D = Molar mass (M) / Molar volume (Vm)
D = 20.328 g/mol / 0.0246153 L/mol
D = 827.85 g/L
Therefore, the density of the breathing mixture at 22 degrees Celsius and 755 mmHg is approximately 827.85 g/L.
To calculate the density of the breathing mixture, we need to first determine the partial pressures of helium (He), nitrogen (N2), and oxygen (O2) in the mixture at the given conditions.
1. Convert the percentage composition of each gas to its molar fraction:
- Helium (He): 34.3% = 0.343
- Nitrogen (N2): 51.7% = 0.517
- Oxygen (O2): 14% = 0.140
2. Convert the molar fraction of each gas to its partial pressure:
- Calculate the total pressure of the mixture using Dalton's law of partial pressures:
Total pressure (P_total) = 755 mmHg
- Calculate the partial pressures of each gas:
Partial pressure of He (P_He) = P_total * molar fraction of He
Partial pressure of N2 (P_N2) = P_total * molar fraction of N2
Partial pressure of O2 (P_O2) = P_total * molar fraction of O2
3. Calculate the number of moles of each gas:
- To find the number of moles, 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 gas constant, and T is the temperature in Kelvin.
- Rearrange the equation to solve for n:
n = PV / RT
4. Find the molar mass of each gas:
- Helium (He): Molar mass = 4.00 g/mol
- Nitrogen (N2): Molar mass = 28.02 g/mol
- Oxygen (O2): Molar mass = 32.00 g/mol
5. Calculate the mass of each gas:
- Mass = number of moles * molar mass
6. Calculate the total mass of the breathing mixture:
- Add the masses of each gas together:
Total mass = mass of He + mass of N2 + mass of O2
7. Calculate the density of the breathing mixture:
- Density = total mass / total volume
Note: Pressure values are generally expressed in units of Pascal (Pa) or atmospheres (atm). In this case, we converted the given pressure value from mmHg to calculate the partial pressures. However, the final density will not depend on the units used as long as temperature and pressure are in consistent units.
Now, follow these steps to calculate the density of the breathing mixture at 22°C and 755 mmHg:
1. Convert the temperature to Kelvin:
Temperature in Kelvin (T) = 22 + 273.15 = 295.15 K
2. Calculate the partial pressures of each gas:
Partial pressure of He (P_He) = 755 mmHg * 0.343
Partial pressure of N2 (P_N2) = 755 mmHg * 0.517
Partial pressure of O2 (P_O2) = 755 mmHg * 0.140
3. Calculate the number of moles of each gas:
n_He = (P_He * 1 atm) / (R * T)
n_N2 = (P_N2 * 1 atm) / (R * T)
n_O2 = (P_O2 * 1 atm) / (R * T)
where R is the ideal gas constant (0.0821 L·atm/mol·K)
4. Find the mass of each gas:
Mass of He = n_He * Molar mass of He
Mass of N2 = n_N2 * Molar mass of N2
Mass of O2 = n_O2 * Molar mass of O2
5. Calculate the total mass of the breathing mixture:
Total mass = Mass of He + Mass of N2 + Mass of O2
6. Determine the volume of the mixture:
Since the volume is not provided, we assume it to be 1 liter (1000 cm^3) for simplicity.
7. Calculate the density:
Density = Total mass / Total volume
By following these steps, you can determine the density of the breathing mixture at 22°C and 755 mmHg.