Question

The $rms$speed of molecules in a gas is $600m/s$. What will be the $rms$ speed if the gas pressure and volume are both halved?

Short answer

The $rms$velocity of volume is $300m/s$.

Step-by-step solution

Definition of gas law 

Gas laws are rules that govern how a gas's pressure, volume, and temperature are related.

The ideal gas law, , is a single generalisation of the behaviour of gases known as an equation of state, where n is the number of gram-moles of a gas and R is the universal gas constant

.$PV=nRT$

Explanation

It's important to know that, according to the ideal gas law,

$pV=nRT$

The heat is proportional to the heat product, in other words. As a result, doubling each of these will give you a temperature that is one-fourth of what it was before.

The formula giving the $rms$speed is

$$\begin{gathered} v_{rms}=\sqrt{\frac{3k_{B}T}{m}} \\ =C\sqrt{T}, \end{gathered}$$

where $C$is a constant. It is therefore clear that if the temperature will become a fourth of the initial one, the $rms$speed will be one half of the initial one. That is, the new speed will be

$$\begin{gathered} v_{\text{new}}=\frac{1}{2} \\ v=\frac{600}{2} \\ =300\mathrm{m}/\mathrm{s} \end{gathered}$$

Bear in mind that this problem can also be answered by looking at two different systems, one liquid-liquid and the other isothermal, and seeing if the formulas for speed as a function of temperature and pressure arrive at the same conclusion.

The rms speed of volume

The $rms$speed of volume is $300m/s$. As the value of temperature and pressure ,respectively.