Consider nitrogen gas, N2, at 27.7°C.
What percentage of nitrogen molecules have a speed within 1.00 m/s of the most probable speed?
Try using these:
https://answers.yahoo.com/question/index?qid=20081123194116AA3FxdL
use that for the average speed and standard deviation.
Then he range is so narrow I would assume a normal distribution and use that average and standard deviation in the normal table rather than trying to integrate the Maxwell probability function or something:
http://davidmlane.com/hyperstat/z_table.html
To find the percentage of nitrogen molecules that have a speed within 1.00 m/s of the most probable speed, we need to use the Maxwell-Boltzmann speed distribution.
The Maxwell-Boltzmann distribution describes the distribution of speeds of particles in a gas at a given temperature. The most probable speed (vmp) in this distribution corresponds to the peak of the curve.
The formula for the most probable speed of a gas particle at a given temperature (T) can be calculated using the root mean square (rms) speed (vrms) and the molar mass (M) of the gas:
vmp = √(2 * Vrms² / 3)
First, let's calculate the most probable speed (vmp) of nitrogen gas at 27.7°C using the root mean square speed (vrms).
The formula for the root mean square speed is given by:
vrms = √(3 * RT / M)
Where R is the gas constant (8.314 J/(mol·K)) and T is the temperature in Kelvin (K).
Converting 27.7°C to Kelvin, we get:
T = 27.7°C + 273.15 = 300.85 K
We also need to know the molar mass of nitrogen gas (N2), which is approximately 28.0134 g/mol.
Now, let's calculate the root mean square speed (vrms):
vrms = √(3 * (8.314 J/(mol·K)) * (300.85 K) / (28.0134 g/mol))
≈ √(2482.23 J/g)
≈ 49.82 m/s
Now that we have the most probable speed (vmp) and its value is 49.82 m/s, we can calculate the range of speeds within 1.00 m/s of the most probable speed.
The range of speeds will be from (vmp - 1.00) m/s to (vmp + 1.00) m/s.
Substituting the values, we get:
(vmp - 1.00) m/s = 49.82 m/s - 1.00 m/s
= 48.82 m/s
(vmp + 1.00) m/s = 49.82 m/s + 1.00 m/s
= 50.82 m/s
Now, we need to find the percentage of nitrogen molecules within this range.
To do that, we need to integrate the Maxwell-Boltzmann speed distribution curve within the range of speeds (vmp - 1.00) m/s to (vmp + 1.00) m/s. The integral represents the probability of finding a molecule within that speed range.
However, integrating the distribution function requires advanced calculus. Instead, we can use the concept that the area under the curve represents the total probability.
Since the Maxwell-Boltzmann speed distribution is a probability density function (PDF) and integrates to 1 over all speeds, the area of the distribution between (vmp - 1.00) m/s and (vmp + 1.00) m/s represents the probability (or percentage) of finding a molecule within that speed range.
So, to find the percentage, we need to find this area under the curve.
To approximate the area under the curve, we can assume that the distribution follows a Gaussian (bell-shaped) curve. In a Gaussian distribution, around 68% of the data falls within one standard deviation of the mean.
Therefore, the percentage of nitrogen molecules with a speed within 1.00 m/s of the most probable speed (vmp) would be approximately 68%.
Therefore, approximately 68% of the nitrogen molecules have a speed within 1.00 m/s of the most probable speed.