Solve these questions in English showing all steps

The given equation is $P = \frac{1}{3} \rho \overline{c^2}$.
The density $\rho$ of a gas is its mass ($m$) divided by its volume ($V$), so $\rho = \frac{m}{V}$.
Substitute this into the pressure equation:

Rearrange to get $PV$:

The total mass of the gas ($m$) can be written as the number of moles ($n$) multiplied by the molar mass ($M$) of the gas: $m = nM$.
Substitute this into the equation from Step 1:

The ideal gas law states $PV = nRT$, where $R$ is the molar gas constant and $T$ is the absolute temperature.
Substitute $nRT$ for $PV$ in the equation from Step 2:

Cancel $n$ from both sides of the equation:

Multiply both sides by 3 and divide by $M$ to isolate $\overline{c^2}$:

This shows the desired relationship.

Initial temperature $T_1 = 30.0^\circ\text{C} = 30.0 + 273.15 = 303.15 \text{ K}$.
Final temperature $T_2 = 150^\circ\text{C} = 150 + 273.15 = 423.15 \text{ K}$.

From part (b), we have $\overline{c^2} = \frac{3RT}{M}$.
The root-mean-square speed ($c_{rms}$) is the square root of the mean-square speed: $c_{rms} = \sqrt{\overline{c^2}} = \sqrt{\frac{3RT}{M}}$.
Since $3R$ and $M$ are constants for a given gas, $c_{rms}$ is directly proportional to the square root of the absolute temperature: $c_{rms} \propto \sqrt{T}$.
Therefore, we can write the ratio:

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

Rounding to three significant figures: