The formula for the Maxwell-Boltzmann distribution is given below,where y is proportional to the fraction of particles of molecular mass m at temperature T traveling at speed u. The expression frac(u1,u2) gives the fraction of particles between speeds of u1 and u2.

Determine the fraction of particles with speeds greater than the root-mean-square speed of the particles at 285.0 K to two significant figures. Be very careful with units and magnitudes here. Note that in this formulation, frac is unitless.

frac(u1,u2)=?

To determine the fraction of particles with speeds greater than the root-mean-square speed at 285.0 K, we need to use the Maxwell-Boltzmann distribution formula and the given expression for the fraction of particles between speeds u1 and u2.

The Maxwell-Boltzmann distribution formula for the fraction of particles with speed u is given by:

f(u) = 4πu^2 * (m / (2πkT))^3/2 * e^(-m*u^2 / (2kT))

Where:
- f(u) is the fraction of particles with speed u
- u is the speed of the particles
- m is the molecular mass of the particles
- k is the Boltzmann constant (1.380649 x 10^-23 J/K)
- T is the temperature in Kelvin
- e is the base of the natural logarithm (approximately 2.71828)

To find the fraction of particles with speeds greater than the root-mean-square speed at 285.0 K, we need to integrate the Maxwell-Boltzmann distribution formula from the root-mean-square speed (urms) to infinity.

1. Calculate the root-mean-square speed (urms) using the formula:

urms = √(3kT / m)

Where:
- urms is the root-mean-square speed
- k is the Boltzmann constant
- T is the temperature in Kelvin
- m is the molecular mass of the particles

2. Integrate the Maxwell-Boltzmann distribution formula from urms to infinity:

frac(u1, u2) = ∫ [4πu^2 * (m / (2πkT))^3/2 * e^(-m*u^2 / (2kT))] du

Where:
- frac(u1, u2) is the fraction of particles between speeds u1 and u2
- u1 is the lower bound of integration (urms)
- u2 is the upper bound of integration (infinity)

By performing this integral, you can determine the fraction of particles with speeds greater than the root-mean-square speed at 285.0 K to two significant figures.