A 500.0 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. 2.15 × 10^{21} molecules
• B. 3.45 × 10^{20} molecules
• C. 1.20 × 10^{23} molecules
• D. 7.98 × 10^{22} 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{500.0}{1000} = 0.5000\, \text{L}$.
2. Convert the temperature from Celsius to Kelvin since the ideal gas law requires temperature in Kelvin. Use 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 (\$0.08206\, \text{L} \cdot \text{atm} / \text{mol} \cdot K\(), and \)T\( is temperature in Kelvin. Rearrange to solve for \)n\(: \)n = \frac{PV}{RT}$.
5. Calculate the number of molecules by multiplying the number of moles \(n\) by Avogadro's number \(N_A = 6.022 \times 10^{23}\, \text{molecules/mol}\). So, \(\text{Number of molecules} = n \times N_A\).