Consider a 50 L tank which hold 20 g of hydrogen gas, 20 g of carbon dioxide gas, and 20 g of krypton gas, at a total pressure of 600 KPa. 1. which gas has the largest partial pressure? 2. Which gas the smallest average speed? 3. What is the temperature of this gas mixture? 4. What is the partial pressure of carbon monoxide? 5. What volume would this gas mixture occupy at STP?
Compute the number of moles n of each gas.
1. The gas with the highest number of moles will have the largest partial pressure.
2. They all have the same T, and average speed is proportional to sqrt(T/M), where M is the molar mass. The lightest gas moved the fastest
3. Use pV = N R T
to compute T
where N is the total number of moles
4. p(CO2)/P = n(CO2)/N
5. V(STP) = 22.4 liters * N
To answer these questions, we can use the ideal gas law equation: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Before we proceed, let's calculate the number of moles for each gas in the tank using their molar masses.
1) To determine which gas has the largest partial pressure, we need to calculate the partial pressures of each gas.
Partial pressure (P_i) = mole fraction (X_i) * total pressure (P_total)
The mole fraction (X_i) can be calculated by dividing the number of moles of the gas (n_i) by the total number of moles of all gases present (n_total).
The mole fraction is given by:
X_i = n_i / n_total
Let's calculate the mole fractions and partial pressures for each gas:
For hydrogen gas:
Molar mass of hydrogen (H2) = 2 g/mol
Number of moles of hydrogen (n_H2) = mass / molar mass = 20 g / 2 g/mol = 10 mol
Number of moles of other gases (n_total) = 10 mol + 10 mol + 10 mol = 30 mol
Mole fraction of hydrogen (X_H2) = n_H2 / n_total = 10 mol / 30 mol = 1/3
Partial pressure of hydrogen (P_H2) = X_H2 * P_total = 1/3 * 600 KPa = 200 KPa
For carbon dioxide gas:
Molar mass of carbon dioxide (CO2) = 44 g/mol
Number of moles of carbon dioxide (n_CO2) = mass / molar mass = 20 g / 44 g/mol = 0.454 mol
Mole fraction of carbon dioxide (X_CO2) = n_CO2 / n_total = 0.454 mol / 30 mol = 0.015
Partial pressure of carbon dioxide (P_CO2) = X_CO2 * P_total = 0.015 * 600 KPa = 9 KPa
For krypton gas:
Molar mass of krypton (Kr) = 83.8 g/mol
Number of moles of krypton (n_Kr) = mass / molar mass = 20 g / 83.8 g/mol = 0.239 mol
Mole fraction of krypton (X_Kr) = n_Kr / n_total = 0.239 mol / 30 mol = 0.008
Partial pressure of krypton (P_Kr) = X_Kr * P_total = 0.008 * 600 KPa = 4.8 KPa
From the above calculations, we can see that hydrogen gas has the largest partial pressure with a value of 200 KPa.
2) To determine which gas has the smallest average speed, we can use the root mean square (rms) speed equation:
rms speed (u) = sqrt(3RT / M)
where R is the ideal gas constant and M is the molar mass of the gas.
For hydrogen gas:
Molar mass of hydrogen (H2) = 2 g/mol
For carbon dioxide gas:
Molar mass of carbon dioxide (CO2) = 44 g/mol
For krypton gas:
Molar mass of krypton (Kr) = 83.8 g/mol
We can calculate the rms speeds for each gas and compare them to find the gas with the smallest average speed.
3) To determine the temperature of the gas mixture, we can rearrange the ideal gas law equation (PV = nRT) to solve for temperature (T):
T = PV / (nR)
Here, we have the total pressure (P_total) and the total number of moles of all gases (n_total). We can substitute these values into the equation and calculate the temperature.
4) To find the partial pressure of carbon monoxide (CO), we need to know the mass or molar fraction of carbon monoxide present in the tank.
5) To calculate the volume the gas mixture would occupy at STP (standard temperature and pressure), we need to use the ideal gas law again. At STP, the temperature is 273 K and the pressure is 1 atm (101.3 kPa). We can rearrange the ideal gas law equation to solve for volume (V):
V = (nRT) / P
For this calculation, we need to know the number of moles of the gas mixture (n_total) and the temperature (T).