At what temperature do hydrogen molecules have the same rms speed as nitrogen molecules at 90 degree C?
1/2 mv^2=1/2 MV^2
M nitrogen is 28, mass Hydrogen= 2
vrms hydrogen = sqrt (28/2) Vrms Nitrogen
At what temperature do hydrogen molecules have the same rms speed as nitrogen molecules at 90 degrees Celsius?
To determine the temperature at which hydrogen molecules have the same root mean square (rms) speed as nitrogen molecules at 90 degrees Celsius, we need to apply the principles of the kinetic theory of gases and the expression for rms speed.
The rms speed of molecules can be calculated using the following equation:
vrms = sqrt((3 * kB * T) / m)
Where:
- vrms is the rms speed of the molecules
- kB is the Boltzmann constant (1.38 x 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molar mass of the molecule in kilograms
First, let's calculate the rms speed of nitrogen molecules at 90 degrees Celsius (363 Kelvin). The molar mass of nitrogen (N2) is approximately 28.0134 grams/mol, which is equivalent to 0.0280134 kg/mol.
vrms_nitrogen = sqrt((3 * kB * T) / m_nitrogen)
= sqrt((3 * 1.38 x 10^-23 J/K * 363 K) / 0.0280134 kg/mol)
Calculating this expression gives us the rms speed of nitrogen molecules at 90 degrees Celsius.
Next, let's determine the temperature at which hydrogen molecules will have the same rms speed as nitrogen at 90 degrees Celsius. The molar mass of hydrogen (H2) is approximately 2.01588 grams/mol, which is equivalent to 0.00201588 kg/mol.
For hydrogen, we use the same equation:
vrms_hydrogen = sqrt((3 * kB * T) / m_hydrogen)
Since we want the rms speed for hydrogen to be the same as for nitrogen at 90 degrees Celsius, we set the two expressions equal to each other and solve for T:
sqrt((3 * 1.38 x 10^-23 J/K * 363 K) / 0.0280134 kg/mol) = sqrt((3 * 1.38 x 10^-23 J/K * T) / 0.00201588 kg/mol)
By squaring both sides and isolating the temperature, we obtain the solution for T.