How many molecules of N_2 are present in a 400.0 mL container at 780 mm Hg and 135°C? (Use R = 0.0821 L·atm·mol^{-1}·K^{-1} and 1 atm = 760 mm Hg.)

• A. 4.02 × 10^{21} molecules
• B. 2.36 × 10^{23} molecules
• C. 7.85 × 10^{22} molecules
• D. 1.20 × 10^{22} molecules

Answer: (C)

Verified step by step guidance

1. Convert the given pressure from mm Hg to atm using the conversion factor: \(P(\text{atm}) = \frac{P(\text{mm Hg})}{760}\). This is necessary because the gas constant \(R\) is given in units involving atm.
2. Convert the temperature from Celsius to Kelvin using the formula: \(T(K) = T(^\circ C) + 273.15\). This is required because the ideal gas law uses absolute temperature.
3. Convert the volume from milliliters to liters since the gas constant \(R\) uses liters: \(V(L) = \frac{V(mL)}{1000}\).
4. Use the ideal gas law equation to find the number of moles of \(N_2\): \( n = \frac{PV}{RT} \) where \(P\) is pressure in atm, \(V\) is volume in liters, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.
5. Calculate the number of molecules by multiplying the number of moles by Avogadro's number: \( \text{Number of molecules} = n \times 6.022 \times 10^{23} \)