To determine how the given speeds compare in an ideal gas, we need to understand the relationships between different types of speeds: the average speed, the most probable speed, the root-mean-square (RMS) speed, and the average vector velocity.
1. Average Speed of Molecules (a): This is the mean speed of all the molecules in the gas sample. It gives us a general sense of how fast the molecules are moving on average.
2. Most Probable Speed of Molecules (b): This is the speed at which the highest number of molecules are moving. In the Maxwell-Boltzmann distribution of speeds, this is the peak of the distribution.
3. Root-Mean-Square (RMS) Speed of Molecules (c): This is a measure of the average speed of molecules, taking into account the square of their velocities. It is higher than the average speed because it places a greater weight on higher velocities.
4. Average Vector Velocity of Molecules (d): For an ideal gas in equilibrium, the velocities of molecules are randomly distributed in all directions, which results in their vector sum (and thus the average vector velocity) being zero.
Let’s rank these speeds from largest to smallest:
- The RMS speed (c) is the highest because it involves squaring the velocities before averaging and then taking the square root, which increases the value. Mathematically, it is always greater than or equal to the average speed.
- The average speed (a) is next. It is the straightforward mean of all molecular speeds without squaring.
- The most probable speed (b), which is the speed that the largest number of molecules likely exhibit, is slightly less than the average speed because it peaks earlier in the Maxwell-Boltzmann distribution.
- The average vector velocity (d), being zero for an ideal gas at equilibrium, is the smallest.
Thus, the ranking from largest to smallest is:
[tex]\[ c > a > b = d \][/tex]
This order takes into account the different statistical measures and properties of molecular speeds in an ideal gas.
