What is the pressure belonging to in Boyle's Law? Just a hour ago, I was reading the great and highly interesting, Boyle's Law. In my textbook, there is given Boyle's Law and leaving some pages, there is stated Dalton's Law of Partial Pressure. 

Boyle's Law is stated as, "As so long as the mass of the gas and temperature is constant, the product of pressure by volume of the gas remains constant."

Experiment of U-tube is well known. I am not going to copy it here but I would directly explain my problem. From U-tube experiment, I grasped that the pressure, mentioned there, is one that is applied and not of the gas. However, the difference stays vague in the law stated earlier that whether this is the pressure applied by external agency or the pressure applied by gas molecules or gas.

Later, I read there the Dalton's Law of partial pressure. There is a primary equation concerning to partial pressure of a gas viz.,

$$\frac {P_{gas}}{P_{mixture}}=\frac{m_{gas}}{M_{mixture}}$$

It is clearly mentioned there that this pressure either $P_{gas}$ or $P_{mixture}$ is the pressure applied by gas/mixture and not the pressure applied by external agency. This is even more clear from the relevant experiment.

The problems start when I begin to analyze the derivation for the equation above. Author of my textbook hasn't given any derivation for it but my teacher let us did that.

Derivation:

Consider  the mixture of gases A, B and C.

Let the number of moles of gas    $A=n_A$

Similarly, number of moles of gas $B=n_B$

And that of C is given by $n_C$

Total number of moles=$n_T=n_A+n_B+n_C$

We know that, $$P=\frac{nRT}{V}$$

Therefore pressure applied by gas A is given by, $\color{red}{P_A=\frac{n_ART}{V}}$

For gas B, $\color{red}{P_B=\frac{n_BRT}{V}}$

And for gas C, $\color{red}{P_C=\frac{n_CRT}{V}}$

For mixture of A, B and C, $\color{red}{P_T=\frac{n_TRT}{V}}$

Diving $P_A$, $P_B$ and $P_C$ individually by $P_T$ we get,

$$\frac{P_A}{P_T}=\frac{n_A}{n_T}$$

$$\frac{P_B}{P_T}=\frac{n_B}{n_T}$$

$$\frac{P_C}{P_T}=\frac{n_C}{n_T}$$


  The red marked statements are where my previous concepts were denounced. I was thinking that the derivation maybe wrong but I am not sure so I want you people to judge my concepts and my teacher as well and inform me.


Thanks.
Your derivation of Dalton's Law of partial pressure is fine. 

I think you are confused about how pressure is "applied" to a gas, and what that does to the internal pressure of a gas. 

Imagine that you have a "sphere" of gas surrounded by a "force field" (like in Star Trek) in empty space, then the force field containing the gas is dropped. Since there is nothing to confine the gas it just expands into infinite space. 

A lot of ideal gas models, like the Wikipedia article on Boyle's Law, show the gas as being contained in a cylinder with a piston pushing on the gas to create the volume and pressure of the system. The external force on the piston is counterbalanced by the pressure of the gas times the surface area of the piston. The pressure of the gas is the same on all surfaces of the container. The gas doesn't just "press back" on the piston. 

Pressure and volume are inversely related for the same system. So at the same temperature, 

$PV = k$

Thus if the pressure increases, then for the same amount of gas the volume must decrease to keep $k$ a constant.