what happen to the speed of gas particles when the pressure increased? There is gas particles inside a piston, if the piston was pushed inward, the volume will decrease and the pressure will increase, so my question is 
What will happen to the speed of the gas particles?
 A: Kinetic theory of gases connects the macroscopic properties $(eg,  P,V,T)$ to the microscopic properties $(eg, v_{rms})$ of a system.  The assumptions of KTG apply to my answer.
Let me first state the symbols that I am going to use and the parameter they represent :
$P$ - Pressure of gas , $V$- Volume of gas , $T$ - Temperature of gas , $n$ - number of moles of gas , $M$ - Mass of gas in V volume , $\rho$ -  density of gas , and $M'$ - Molecular mass of gas. 
According to KTG,
$P=\frac{\rho v_{rms}^{2}}{3}$.
$v_{rms}^{2}=\frac{3PV}{M}$ , 
Using ideal gas equation, $PV=nRT$
$v_{rms}^{2}=\frac{3RT}{M'}$ , beacuse $M=nM'$.
As you can see,
$v_{rms}\propto \sqrt{T}$.
The change is speed of gas particles actually depends on the type of process the system undergoes.
We can have two cases :
1) When $\Delta T =0$:
If $\Delta {T}=0$ (isothermal process, $P \propto \frac{1}{V}$), then there will be no change in the speed the of the gas particles.
In this case when the gas is compressed to half the volume, the pressure is doubled. Therefore for such case, speed of the gas particles remains the same.
2) When $\Delta T \neq 0$ :
If $\Delta T \neq 0 $ (Say, adiabatic process), then the speed of the particles will increase with increase in temperature and vice versa. Adiabatic expansion cools the system and adiabatic compression heats up the system.
Similarly, we can make comparisons , For :
Isobaric processes - $\Delta P=0, V\propto T \propto v_{rms}^{2}$ and
isochoric processes - $\Delta V =0,P\propto T \propto v_{rms}^{2}$.
Conclusion : A general comment cannot be made. The process that the system is undergoing has to be mentioned.
A: If the system is thermally isolated then the pressure and the temperature of the gas will increase.
The reason for this is that as the piston moves inward the rebound speed of the molecules will be larger than the speed before rebound.
After the molecules have collided with others this produces an increase in the overall kinetic energy due to the random motion of the molecules i.e. the gas temperature has increased.
The pressure has increased because the rate of collision between the molecules and the walls has increased and the change in momentum of a molecule when it rebounds has also increased.  
If the system has walls which are at a constant temperature then on average the kinetic energy of the molecules does not change but the rate at which the molecules hit the walls increases which produces an increase in pressure.
