Calculate the root mean square,average and most probable speed of oxygen at 27degree celcius

Please type your subject in the School Subject box. Any other words, including obscure abbreviations, are likely to delay responses from a teacher who knows that subject well.

VRMS =sqrt[(3RT)/M]

and The most probable speed is 81.6% of the rms speed, and the average speed 92.1% (distribution of speeds).

Take care with units of M and R

To calculate the root mean square (rms), average, and most probable speed of oxygen at a given temperature, we need to use the Maxwell-Boltzmann speed distribution equation. The equation is as follows:

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

Where:
f(v) is the probability density function of finding a particle with speed v
v is the speed of the particle
m is the mass of the particle
k is the Boltzmann constant (1.38 × 10^-23 J/K)
T is the temperature in Kelvin

To start, we need to convert the given temperature of 27 degrees Celsius to Kelvin. The formula to convert Celsius to Kelvin is:

T(K) = T(°C) + 273.15

So, let's convert the temperature:
T(K) = 27 + 273.15 = 300.15 K

Now, let's calculate the root mean square (rms) speed:
To find the rms speed, we need to integrate the speed distribution equation from 0 to infinity and then solve for the root mean square (rms) speed. However, we can simplify this by using the equation:

v(rms) = √(3kT / m)

Let's substitute the values:
v(rms) = √((3 × 1.38 × 10^-23 J/K × 300.15 K) / (32 × 1.66 × 10^-27 kg))

Using a calculator, we get:
v(rms) ≈ 484.2 m/s (rounded to three significant figures)

Next, let's calculate the average speed:
The average speed of particles can be found by integrating the speed distribution equation from 0 to infinity and then solving for the average speed. However, we can simplify this by using the equation:

v(avg) = √(8kT / πm)

Let's substitute the values:
v(avg) = √((8 × 1.38 × 10^-23 J/K × 300.15 K) / (π × 32 × 1.66 × 10^-27 kg))

Using a calculator, we get:
v(avg) ≈ 533.1 m/s (rounded to three significant figures)

Lastly, let's calculate the most probable speed:
The most probable speed can be obtained by finding the maximum value of the speed distribution function and substituting it into the equation. The maximum value occurs when the derivative of the speed distribution function is equal to zero.

Differentiating the speed distribution equation with respect to v and setting it to zero, we get:
df(v) / dv = (4πv²) × (m / 2πkT)^(3/2) × exp(-mv² / 2kT) × (-mv / kT) + (4πv) × (m / 2πkT)^(3/2) × exp(-mv² / 2kT) = 0

Simplifying the equation, we get:
- v + 2v = 0
v = 2v

So, the most probable speed (vmp) is equal to twice the root mean square (rms) speed. Hence:
vmp = 2 × v(rms)

Substituting the value of v(rms) we calculated earlier, we have:
vmp = 2 × 484.2 m/s = 968.4 m/s (rounded to three significant figures)

Therefore, at 27 degrees Celsius, the root mean square (rms) speed of oxygen is approximately 484.2 m/s, the average speed is approximately 533.1 m/s, and the most probable speed is approximately 968.4 m/s.