Calculating the Temperature at which the Root Mean Square Velocity of Hydrogen Molecules will Exceed their Most Probable Velocity by 400 m/s

To calculate the temperature at which the root mean square (RMS) velocity of hydrogen molecules exceeds their most probable velocity by 400 m/s, we need to consider the Maxwell-Boltzmann distribution of molecular speeds.

Maxwell-Boltzmann Distribution
The Maxwell-Boltzmann distribution describes the distribution of molecular speeds in a gas at a given temperature. It is given by the equation:

f(v) = 4πv² * (m / 2πkT)^(3/2) * exp(-mv² / 2kT)

Where:
- f(v) is the fraction of molecules with speed v
- v is the molecular speed
- m is the mass of a molecule
- k is the Boltzmann constant
- T is the absolute temperature

Most Probable Velocity
The most probable velocity (vmp) is the speed at which the fraction of molecules with that speed is maximum. It can be found by differentiating the Maxwell-Boltzmann distribution equation with respect to v and setting it equal to zero:

df(v) / dv = 0

Differentiating the equation and solving for v, we find that the most probable velocity is:

vmp = sqrt(2kT / m)

Root Mean Square Velocity
The root mean square velocity (vrms) is the square root of the mean of the squares of the velocities. It can be calculated by finding the average value of v² and taking the square root:

vrms = sqrt(3kT / m)

Calculating the Temperature
To find the temperature at which the RMS velocity exceeds the most probable velocity by 400 m/s, we can set up the following equation:

vrms - vmp = 400

Substituting the expressions for vrms and vmp:

sqrt(3kT / m) - sqrt(2kT / m) = 400

Squaring both sides of the equation and isolating T, we can solve for the temperature:

3kT / m - 2kT / m - 2 * sqrt(6k²T² / m²) = 160000

Simplifying the equation:

kT / m - 2 * sqrt(6k²T² / m²) = 160000

2 * sqrt(6k²T² / m²) = kT / m - 160000

Squaring both sides again:

24k²T² / m² = (kT / m - 160000)²

Simplifying and rearranging:

24k²T² / m² = (kT / m)² - 320000T + 25600000000

Multiplying through by m²:

24k²T² = k²T² - 320000mkT + 25600000000m²

Canceling out the k²T² term:

24k²T² - k²T² = 25600000000m² - 320000mkT

Simplifying and rearranging:

23k²T² = 25600000000m² - 320000mkT

Dividing