Question

What new temperature will a gas have if its speed has to become doubled compared to `127^@ "C"`?

Answer 1

Whatever quadruple the temperature in `"K"` is. Would this apply to RMS speed and average speed as well? Why?

---

The most probable speed is given by

`v_(mp) = sqrt((2k_BT)/m)`,

where:

• `k_B` is the Boltzmann constant in `"J/K"`.
• `T` is temperature in `"K"`.
• `m` is the single-particle mass in `"kg"`.

If you want to double the most probable speed as that at `127^@ "C"`, or `"400.15 K"`, you want...

`v_(mp)' = 2v_(mp) = 2sqrt((2k_BT)/m)`

`= sqrt((2k_B cdot color(red)(4T))/m)`

i.e. the temperature in `"K"` should quadruple to `color(blue)("1600.60 K")`.

---

Or with any other speed that follows from the Maxwell-Boltzmann distribution, i.e. any other speed that has the relationship:

`v prop sqrtT`

We have...

`2v prop sqrt(4T)`

`=>` temperature in `"K"` must have quadrupled.