Consider hydrogen gas, H2, at 24.3°C.
What percentage of hydrogen molecules have a speed within 1.00 m/s of the most probable speed?
To find the percentage of hydrogen 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 speed distribution describes the probability of finding gas molecules at different speeds at a given temperature. It is given by the equation:
f(v) = (4πv^2 / (π^(1/2) (v_mostProbable^3) )) * exp(-(v/v_mostProbable)^2)
Where:
- f(v) is the probability density function at a given speed v.
- v is the speed.
- v_mostProbable is the most probable speed, which can be calculated as (2kT / m)^0.5.
- k is the Boltzmann constant (1.380649 × 10^-23 J/K).
- T is the temperature in Kelvin.
- m is the molar mass of hydrogen (2 g/mol).
To find the percentage of molecules with a speed within 1.00 m/s of the most probable speed, we need to integrate the probability density function from (v_mostProbable - 1.00) m/s to (v_mostProbable + 1.00) m/s and then multiply by 100.
Now let's calculate it step by step:
1. Convert the temperature from Celsius to Kelvin:
T = 24.3 + 273.15 = 297.45 K
2. Calculate the most probable speed, v_mostProbable:
v_mostProbable = (2kT / m)^0.5
= (2 * (1.380649 × 10^-23 J/K) * (297.45 K) / (2 g/mol))^0.5
3. Calculate the range for integration:
lower limit = v_mostProbable - 1.00 m/s
upper limit = v_mostProbable + 1.00 m/s
4. Calculate the value of the probability density function at the lower and upper limits:
f_lower = (4π(lower limit)^2 / (π^(1/2) (v_mostProbable^3) )) * exp(-(lower limit/v_mostProbable)^2)
f_upper = (4π(upper limit)^2 / (π^(1/2) (v_mostProbable^3) )) * exp(-(upper limit/v_mostProbable)^2)
5. Integrate the probability density function from the lower limit to the upper limit:
integral = ∫[lower limit to upper limit] (4πv^2 / (π^(1/2) (v_mostProbable^3) )) * exp(-(v/v_mostProbable)^2) dv.
You can use numerical integration techniques or software to evaluate the integral.
6. Multiply the result by 100 to get the percentage of molecules:
percentage = integral * 100.
By following these steps, you should be able to calculate the percentage of hydrogen molecules that have a speed within 1.00 m/s of the most probable speed.