At what temperature do the molecules of nitrogen gas have an rms speed of 34 m/s?

34 m/s is a typical highway speed for a car.

The root mean square (rms) speed of molecules is given by the equation:

v = √(3kT/m)

Where:
v is the rms speed of the molecules,
k is the Boltzmann constant (1.38 × 10^(-23) J/K),
T is the temperature in Kelvin, and
m is the molar mass of the gas.

For nitrogen gas (N₂), the molar mass is approximately 28.02 g/mol.

Given that the rms speed is 34 m/s, we can rearrange the equation to solve for the temperature (T):

T = (v² * m) / (3k)

Plugging in the given values:
v = 34 m/s
m = 28.02 g/mol
k = 1.38 × 10^(-23) J/K

Let's calculate it:

T = (34² * 0.02802) / (3 * 1.38 × 10^(-23))

T ≈ 2.06 × 10^5 K (Kelvin)

Therefore, at approximately 2.06 × 10^5 K, the molecules of nitrogen gas will have an rms speed of 34 m/s.

To determine the temperature at which molecules of nitrogen gas have an rms (root mean square) speed of 34 m/s, we can use the equation for the rms speed:

v_rms = √(3kT/m)

Where:
- v_rms is the rms speed of the gas molecules
- k is the Boltzmann constant (1.38 x 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molar mass of the gas (for nitrogen gas, m = 28.0134 g/mol)

Given that v_rms = 34 m/s, we can rearrange the equation to solve for T:

T = (v_rms^2 * m) / (3k)

Plugging in the known values:

T = (34^2 * 28.0134) / (3 * 1.38 x 10^-23)

Calculating this expression will give us the temperature in Kelvin:

T ≈ 288.58 K

Therefore, the temperature at which the molecules of nitrogen gas have an rms speed of 34 m/s is approximately 288.58 Kelvin.