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A binary mixture of oxygen and nitrogen with partial pressures in the ratio 0.21 and 0.79 is contained in a vessel at 300 K. If the total pressure of the mixture is 1 * 10 5 N/m2, find the average molecular weight of the mixture
- a)28.84
- b)29.84
- c)30.84
- d)31.84
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A binary mixture of oxygen and nitrogen with partial pressures in the ...
Explanation: (0.21) (32) + (0.79) (28) = 28.84.
Most Upvoted Answer
A binary mixture of oxygen and nitrogen with partial pressures in the ...
Given information:
Partial pressure of oxygen: P_O2 = 0.21 * (1 * 10^5) N/m^2
Partial pressure of nitrogen: P_N2 = 0.79 * (1 * 10^5) N/m^2
Total pressure of the mixture: P_total = 1 * 10^5 N/m^2
Temperature: T = 300 K
We can use the ideal gas law to relate the partial pressures of oxygen and nitrogen to their respective number of moles. The ideal gas law is given by:
PV = nRT
Where:
P is the pressure of the gas (in N/m^2)
V is the volume of the gas (in m^3)
n is the number of moles of the gas
R is the ideal gas constant (8.314 J/mol·K)
T is the temperature of the gas (in K)
Since we have a mixture of oxygen and nitrogen, the total pressure can be expressed as the sum of the partial pressures:
P_total = P_O2 + P_N2
We can rearrange the equation to solve for the number of moles of each gas:
n_O2 = (P_O2 * V) / (RT)
n_N2 = (P_N2 * V) / (RT)
The average molecular weight of the mixture can be calculated by taking the weighted average of the molecular weights of oxygen and nitrogen:
M_avg = (n_O2 * M_O2 + n_N2 * M_N2) / (n_O2 + n_N2)
Where:
M_O2 is the molecular weight of oxygen
M_N2 is the molecular weight of nitrogen
Let's calculate the average molecular weight.
Calculation:
1. Calculate the number of moles of oxygen and nitrogen:
n_O2 = (P_O2 * V) / (RT)
n_N2 = (P_N2 * V) / (RT)
2. Calculate the average molecular weight:
M_avg = (n_O2 * M_O2 + n_N2 * M_N2) / (n_O2 + n_N2)
Substituting the values:
M_avg = ((P_O2 * V) / (RT) * M_O2 + (P_N2 * V) / (RT) * M_N2) / ((P_O2 * V) / (RT) + (P_N2 * V) / (RT))
Since the volume (V), gas constant (R), and temperature (T) are constant, they cancel out:
M_avg = (P_O2 * M_O2 + P_N2 * M_N2) / (P_O2 + P_N2)
Substituting the given values:
M_avg = (0.21 * (1 * 10^5) * M_O2 + 0.79 * (1 * 10^5) * M_N2) / ((0.21 * (1 * 10^5)) + (0.79 * (1 * 10^5)))
Simplifying:
M_avg = (0.21 * M_O2 + 0.79 * M_N2) / (0.21 + 0.79)
M_avg = (0.21 * M_O2 + 0.79 * M_N2) / 1
M_avg = 0.21
Partial pressure of oxygen: P_O2 = 0.21 * (1 * 10^5) N/m^2
Partial pressure of nitrogen: P_N2 = 0.79 * (1 * 10^5) N/m^2
Total pressure of the mixture: P_total = 1 * 10^5 N/m^2
Temperature: T = 300 K
We can use the ideal gas law to relate the partial pressures of oxygen and nitrogen to their respective number of moles. The ideal gas law is given by:
PV = nRT
Where:
P is the pressure of the gas (in N/m^2)
V is the volume of the gas (in m^3)
n is the number of moles of the gas
R is the ideal gas constant (8.314 J/mol·K)
T is the temperature of the gas (in K)
Since we have a mixture of oxygen and nitrogen, the total pressure can be expressed as the sum of the partial pressures:
P_total = P_O2 + P_N2
We can rearrange the equation to solve for the number of moles of each gas:
n_O2 = (P_O2 * V) / (RT)
n_N2 = (P_N2 * V) / (RT)
The average molecular weight of the mixture can be calculated by taking the weighted average of the molecular weights of oxygen and nitrogen:
M_avg = (n_O2 * M_O2 + n_N2 * M_N2) / (n_O2 + n_N2)
Where:
M_O2 is the molecular weight of oxygen
M_N2 is the molecular weight of nitrogen
Let's calculate the average molecular weight.
Calculation:
1. Calculate the number of moles of oxygen and nitrogen:
n_O2 = (P_O2 * V) / (RT)
n_N2 = (P_N2 * V) / (RT)
2. Calculate the average molecular weight:
M_avg = (n_O2 * M_O2 + n_N2 * M_N2) / (n_O2 + n_N2)
Substituting the values:
M_avg = ((P_O2 * V) / (RT) * M_O2 + (P_N2 * V) / (RT) * M_N2) / ((P_O2 * V) / (RT) + (P_N2 * V) / (RT))
Since the volume (V), gas constant (R), and temperature (T) are constant, they cancel out:
M_avg = (P_O2 * M_O2 + P_N2 * M_N2) / (P_O2 + P_N2)
Substituting the given values:
M_avg = (0.21 * (1 * 10^5) * M_O2 + 0.79 * (1 * 10^5) * M_N2) / ((0.21 * (1 * 10^5)) + (0.79 * (1 * 10^5)))
Simplifying:
M_avg = (0.21 * M_O2 + 0.79 * M_N2) / (0.21 + 0.79)
M_avg = (0.21 * M_O2 + 0.79 * M_N2) / 1
M_avg = 0.21
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