You have a 200.0 L vessel at 200.0 K that contains 15.0 g of CH4 gas and 15.0 g of CO gas, what is the total pressure of the system?
The correct answer and explanation is:
To calculate the total pressure of the system, we can use the ideal gas law for each gas individually and then add their pressures together, since the gases are non-reactive and behave ideally. The ideal gas law is:
PV=nRTPV = nRT
Where:
- PP = pressure (in atmospheres)
- VV = volume (in liters)
- nn = number of moles of gas
- RR = ideal gas constant = 0.0821 L·atm/(mol·K)
- TT = temperature (in Kelvin)
Step 1: Calculate moles of CH₄ and CO
To find the number of moles, we use the formula: n=massmolar massn = \frac{\text{mass}}{\text{molar mass}}
For CH₄ (methane):
- Molar mass of CH₄ = 16.04 g/mol
- Moles of CH₄ = 15.0 g16.04 g/mol=0.935 mol\frac{15.0 \, \text{g}}{16.04 \, \text{g/mol}} = 0.935 \, \text{mol}
For CO (carbon monoxide):
- Molar mass of CO = 28.01 g/mol
- Moles of CO = 15.0 g28.01 g/mol=0.536 mol\frac{15.0 \, \text{g}}{28.01 \, \text{g/mol}} = 0.536 \, \text{mol}
Step 2: Calculate the partial pressures
Now, use the ideal gas law for each gas:
For CH₄: PCH₄=nRTV=(0.935 mol)(0.0821 L\cdotpatm/mol\cdotpK)(200.0 K)200.0 LP_{\text{CH₄}} = \frac{nRT}{V} = \frac{(0.935 \, \text{mol})(0.0821 \, \text{L·atm/mol·K})(200.0 \, \text{K})}{200.0 \, \text{L}} PCH₄=15.3200.0=0.0765 atmP_{\text{CH₄}} = \frac{15.3}{200.0} = 0.0765 \, \text{atm}
For CO: PCO=nRTV=(0.536 mol)(0.0821 L\cdotpatm/mol\cdotpK)(200.0 K)200.0 LP_{\text{CO}} = \frac{nRT}{V} = \frac{(0.536 \, \text{mol})(0.0821 \, \text{L·atm/mol·K})(200.0 \, \text{K})}{200.0 \, \text{L}} PCO=8.8200.0=0.0440 atmP_{\text{CO}} = \frac{8.8}{200.0} = 0.0440 \, \text{atm}
Step 3: Total pressure
The total pressure is the sum of the partial pressures of the two gases: Ptotal=PCH₄+PCO=0.0765 atm+0.0440 atm=0.1205 atmP_{\text{total}} = P_{\text{CH₄}} + P_{\text{CO}} = 0.0765 \, \text{atm} + 0.0440 \, \text{atm} = 0.1205 \, \text{atm}
Final Answer:
The total pressure of the system is 0.1205 atm.
This calculation assumes ideal gas behavior, which is a good approximation at the given conditions (200.0 K and 200.0 L vessel). If the gases were interacting or deviating from ideal behavior, corrections would be necessary.