i need the full solutions of the integrals that lead to these equations:
mean speed of molecules in a gas:
vavg= sqrt(8RT/PiM)
average of the square of speeds:
(v^2)avg= 3RT/M
fraction of molecules with speed:
4PI*(M/2PiRT)^(3/2)*e^(-Mv^2/RT)(deltav)
the general integral given is integral from 0 to infinity of v*P(v) where P(v)=4Pi*(M/2PiRT)^(3/2)*v^2*e^(-Mv^2/RT)
Look for this classic book in your school library. It used to be available in paperback:
http://www.amazon.com/Mathematical-Theory-Non-uniform-Gases-Conduction/dp/052140844X
To find the full solutions of these integrals, let's go through each equation step by step:
1. Mean speed of molecules in a gas:
The first equation gives you the mean speed of molecules in a gas. To find the integral that leads to this equation, we need to calculate the average value of the speed of molecules.
The integral you mentioned is:
∫[0 to ∞] v * P(v) dv
Here, P(v) represents the probability density function of the speed distribution, which can be expressed as:
P(v) = 4π * (M/2πRT)^(3/2) * v^2 * exp(-Mv^2/RT)
To calculate the integral, you can use the following steps:
1. Multiply v with P(v) to get v * P(v).
2. Substitute P(v) with its expression.
3. Integrate the resulting expression from 0 to infinity with respect to v.
After performing this integration, you will obtain the equation for the mean speed of molecules in a gas:
v_avg = sqrt(8RT/πM)
2. Average of the square of speeds:
The second equation gives you the average of the square of speeds. To find the integral that leads to this equation, we need to calculate the average value of v^2.
Similarly to the previous integral, the integral you mentioned is:
∫[0 to ∞] v^2 * P(v) dv
Using the same steps as before, you can evaluate this integral to obtain the equation for the average of the square of speeds:
(v^2)_avg = 3RT/M
3. Fraction of molecules with speed:
The third equation gives you the fraction of molecules with a specific speed v. This equation involves the calculation of the probability distribution function.
The integral you mentioned is:
∫[v-dv/2 to v+dv/2] P(v) dv
To calculate this integral, you can substitute P(v) with its expression and perform the integration. This will give you the equation for the fraction of molecules with a specific speed.
4. General integral:
The last equation you mentioned is a general integral, which is the integral of v * P(v) from 0 to infinity. This integral represents the average value of v times the probability density function P(v).
By following the steps mentioned previously, you can calculate this integral to obtain the desired equation.
Please note that the provided equations are derived from statistical mechanics and the Maxwell-Boltzmann distribution, assuming an ideal gas. The derived results may vary depending on the assumptions and conditions used.