Final answer:
The temperature of the nitrogen gas is approximately 549.81 kelvins. Using conversions and the ideal gas law, the pressure was converted to atm, and the number of moles was calculated with the molar mass of nitrogen to solve for temperature.
Step-by-step explanation:
To find the temperature of the nitrogen gas in kelvins, we use the ideal gas law, 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 in kelvins.
First, we convert the pressure from mmHg to atm: 680 mmHg * (1 atm / 760 mmHg) = 0.895 atm. Next, we calculate the number of moles using the molar mass of N₂, which is 28.01 g/mol: 27.8 g / 28.01 g/mol = 0.992 moles.
Now, we can rearrange the ideal gas law to solve for T: T = (PV) / (nR). Substituting the values we have, T = (0.895 atm * 50.0 L) / (0.992 mol * 0.0821 L.atm.mol⁻¹.K⁻¹) = (44.75 L.atm) / (0.0813 mol.K), which gives us T ≈ 549.81 K.