Calculating most probable speed

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In summary, to calculate the most probable speed of diatomic carbon dioxide gas with a molar mass of 44.0 g/mol at a temperature of 305 K, we can use the equation v_mp = sqrt(2kT/m). Substituting in the values and units, we get v_mp = 339 m/s. However, this answer seems low for the speed of molecules, so further math may be needed to verify the result.
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Homework Statement



Diatomic carbon dioxide gas (CO2 has molar mass of 44.0 g/mol) is at a temperature of 305 K.

Calculate the most probable speed v_mp.

Homework Equations



v_mp = sqrt(2kT/m)


The Attempt at a Solution



v_mp = sqrt(2 * 1.3801 * 10^-23 * 305 / 44)

v_mp = 1.383 * 10^-11
 
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  • #2
How about you write down the units of all the numerical values you substituted? It is always good practice to do this, and cancel out the units.
 
  • #3
I can see that m is not simply 44 mol. It needs to be in kg. m = M/N_A. The units for M is g/mol. What are the units for N_A, Avogadro's number?
 
  • #4
well, it's a number... :smile:

6.023E23 particles/mole
 
  • #5
Then

v_mp = sqrt(2 * 1.3801 * 10^-23 (m^2 * kg / (s^2 * K) * 305 K / (44 g/mol / (6.02E23 particles/mol) * kg/1000g)

This ends up with the units being sqrt(m^2/s^2 * particles)

Is that correct?
 
  • #6
heh, well particles isn't really a physical unit (neither is mol). So what you really have is sqrt(m^2/s^2).
 
  • #7
Got it. The answer was 339 m/s. Thanks!
 
Last edited:
  • #8
Looks like molecules should be moving a little faster than that. Check your math. The approach is correct, however.
 

1. What is the most probable speed?

The most probable speed refers to the speed at which the majority of particles in a gas or liquid are moving. It is the most likely speed that a particle will have at a given temperature.

2. How is the most probable speed calculated?

The most probable speed can be calculated using the Maxwell-Boltzmann distribution, which takes into account the mass of the particles and the temperature of the substance. The formula is vmp = √(2RT/M), where vmp is the most probable speed, R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the particles.

3. What factors affect the most probable speed?

The most probable speed is affected by the temperature and mass of the particles. As the temperature increases, the most probable speed also increases. Additionally, lighter particles have a higher most probable speed compared to heavier particles at the same temperature.

4. How does the most probable speed relate to the average speed?

The most probable speed is the speed at which the majority of particles are moving, while the average speed is the overall average of all the particles' speeds. The most probable speed is typically higher than the average speed, as there are a few particles with very high speeds that bring up the average.

5. Can the most probable speed be greater than the speed of light?

No, the most probable speed cannot be greater than the speed of light, as the speed of light is the maximum speed at which any particle can travel according to the theory of relativity. The most probable speed is typically much lower than the speed of light, even at high temperatures.

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