At what temperature is the root mean square speed of oxygen molecules equal to the root mean square speed of hydrogen at 27 degrees?

To find the temperature at which the root mean square (rms) speed of oxygen molecules is equal to the rms speed of hydrogen at 27 degrees Celsius, we can use the equation:

rms speed = sqrt((3 * k * T) / m)

Where:
- rms speed is the root mean square speed
- k is the Boltzmann constant (1.38 * 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molar mass of the gas in kilograms

First, let's calculate the rms speed for hydrogen at 27 degrees Celsius (300 Kelvin). The molar mass of hydrogen (H2) is approximately 2 g/mol, which is 0.002 kg/mol.

rms speed (hydrogen) = sqrt((3 * k * T) / m)
= sqrt((3 * 1.38 * 10^-23 * 300) / 0.002)

Now, calculate the rms speed for oxygen. The molar mass of oxygen (O2) is approximately 32 g/mol, which is 0.032 kg/mol.

rms speed (oxygen) = sqrt((3 * k * T) / m)
= sqrt((3 * 1.38 * 10^-23 * T) / 0.032)

To find the temperature at which the rms speeds of hydrogen and oxygen are equal, we equate the two equations:

sqrt((3 * 1.38 * 10^-23 * 300) / 0.002) = sqrt((3 * 1.38 * 10^-23 * T) / 0.032)

Solve this equation for T:

(3 * 1.38 * 10^-23 * 300) / 0.002 = (3 * 1.38 * 10^-23 * T) / 0.032

Simplifying this equation gives:

900 / 0.002 = T / (0.032 / 0.002)

T = (900 * 0.032) / 0.002

Therefore, the temperature at which the rms speed of oxygen molecules is equal to the rms speed of hydrogen at 27 degrees Celsius is:

T = (900 * 0.032) / 0.002 = 14,400 K

So, the answer is 14,400 Kelvin.