A 250.0 mL container holds nitrogen gas (N_2) at a pressure of 780 mm Hg and a temperature of 135 °C. Using the ideal gas law, how many molecules of N_2 are present in the container?

• A. 6.02 × 10^{23} molecules
• B. 1.02 × 10^{22} molecules
• C. 2.11 × 10^{20} molecules
• D. 4.38 × 10^{21} molecules

Answer: (D)

Verified step by step guidance

1. Convert the given volume from milliliters to liters because the ideal gas law requires volume in liters. Use the conversion: \$1\, \text{L} = 1000\, \text{mL}\(. So, \)V = \frac{250.0}{1000} = 0.2500\, \text{L}$.
2. Convert the temperature from degrees Celsius to Kelvin using the formula: \(T(K) = T(^\circ C) + 273.15\). So, \(T = 135 + 273.15 = 408.15\, K\).
3. Convert the pressure from mm Hg to atmospheres because the ideal gas constant \(R\) is typically given in units involving atmospheres. Use the conversion: \$1\, \text{atm} = 760\, \text{mm Hg}\(. So, \)P = \frac{780}{760} = 1.0263\, \text{atm}$.
4. Use the ideal gas law equation: \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin. Rearrange to solve for \(n\): \(n = \frac{PV}{RT}\). Use \(R = 0.0821\, \text{L} \cdot \text{atm} / \text{mol} \cdot K\).
5. Calculate the number of molecules by multiplying the number of moles \(n\) by Avogadro's number \(N_A = 6.022 \times 10^{23}\) molecules/mol: \(\text{Number of molecules} = n \times N_A\).