A gas bulb of 1 litre capacity contains 2.0 xx 10^(21) molecules of nitrogen exerting a pressure of 7.57 xx 10^3 Nm^(-2) . Calculate the root mean square speed and the temperature of gas molecules. If the ratio of most probable speed to the root mean square speed is 0.82, calculate the most probable speed for the molecules at this temperature.

A gas bulb of 1 litre capacity contains 2.0 xx 10^(21) molecules of ni

1.	A gas bulb of 1 litre capacity contains 2.0 xx 10^(21) molecules of nitrogen exerting a pressure of 7.57 xx 10^3 Nm^(-2) . Calculate the root mean square speed and the temperature of gas molecules. If the ratio of most probable speed to the root mean square speed is 0.82, calculate the most probable speed for the molecules at this temperature.
Answer» Solution :Number of moles of NITROGEN = `("number of molecules")/(6.023 xx 10^(23))` `= (2.0 xx 10^(11))/(6.023 xx 10^(23)) = 3.32 xx 10^(-3)` According to the gas equation PV = NRT In the PRESENT case, `P = 7.57 xx 10^3 N m^(-2), V = 1` `L = 10^(-3) m^(3)`, `R = 8.314 J K^(-1) mol^(-1)` T = ? Substituting the values, we have `7.57 xx 10^3 xx 10^(-3) = 3.32 xx 10^(-3) xx 8.314 XXT` `:. "" T= 274.2 K` RMS speed is given by `u = sqrt((3RT)/M)= sqrt((3 xx 8.314 xx 274.2)/(28 xx 10^(-3)))= 494.2 m s^(-1)` (Molar mass of `N_2 " in kg " = 28 xx 10^(-3)`) `:. " Most probable speed " = 0.82 xx RMS` speed `:. " Most probable speed of " N_2` under given conditions `= 0.82 xx 494.2 = 405.2 m s^(-1)`