Solve showing all the steps, in English

Density is mass per unit volume, $\rho = \frac{m}{V}$.
The total mass of the gas ($m$) can also be expressed as the number of moles ($n$) multiplied by the molar mass ($M$), so $m = nM$.
Therefore, $\rho = \frac{nM}{V}$.

Equate the expression for $PV$ from Step 4 with the Ideal Gas Law:

Multiply both sides by 3 and divide by $M$:

This shows the relationship between the mean-square speed ($c^2$) and the absolute temperature ($T$).

Initial temperature, $T_1 = 30.0 \text{ °C} + 273.15 = 303.15 \text{ K}$
Final temperature, $T_2 = 150 \text{ °C} + 273.15 = 423.15 \text{ K}$

From part (b), we know that $c^2 = \frac{3RT}{M}$.
This implies that the mean-square speed ($c^2$) is directly proportional to the absolute temperature ($T$).
Therefore, the root-mean-square speed ($c_{rms} = \sqrt{c^2}$) is proportional to the square root of the absolute temperature:

So, we can write the ratio:

Given initial rms speed, $c_{rms,1} = 850 \text{ m s}^{-1}$.

The root-mean-square speed of the molecules if the gas is heated to $150 \text{ °C}$ is $\boxed{1,00 \times 10^3 \text{ m s}^{-1}}$.