At a certain temperature, hydrogen molecules have r.m.s. velocity of 3 km/s. what is the r.m.s velocity of the oxygen molecules at the same temperature? (a) 0.25 km/s (b) 0.5 km/s (c) 0.75 km/s (d) 6 km/s

To solve this problem, we use the formula for the root mean square (r.m.s) velocity of gas molecules:

v_rms = sqrt( (3RT) / M )

where:

v_rms is the root mean square velocity,
R is the universal gas constant,
T is the temperature,
M is the molar mass of the gas.
Since the temperature is the same for both gases (hydrogen and oxygen), we take the ratio of their r.m.s. velocities:

v_rms(H2) / v_rms(O2) = sqrt( M(O2) / M(H2) )

Given that:

The molar mass of hydrogen (H2) is 2 g/mol,
The molar mass of oxygen (O2) is 32 g/mol,
The r.m.s. velocity of hydrogen molecules is given as 3 km/s.
Now, substituting these values:

v_rms(O2) = v_rms(H2) * sqrt( M(H2) / M(O2) )
= 3 km/s * sqrt( 2 / 32 )
= 3 km/s * sqrt(1/16)
= 3 km/s * (1/4)
= 0.75 km/s

Thus, the correct answer is 0.75 km/s, which corresponds to option (c).