How do I prove
$${{\displaystyle p_{i}=p_{\text{total}}y_{i}}}$$
where $y_i$ is the mole fraction of the $i^{th}$ component in the total mixture of $n$ components ?
Here is what I know
Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation:
${\displaystyle p_{\text{total}}=\sum _{i=1}^{n}p_{i}}$
or
${\displaystyle p_{\text{total}}=p_{1}+p_{2}+p_{3}+...p_{n}}$
And mole fraction $y_i= \dfrac{n_i}{n_{\text {total}}}$
I don't see how to relate , please help.
After having some discussion with @JohnRennie
Let $P_i=n_i \dfrac{RT}{V}$ then $P_i=\dfrac{n_i}{n_{\text{total}}}\dfrac{n_{\text{total}}RT}V$
$P_i=x_i P_{\text{total}}$
Where $x_i$ is mole fraction of $i^{\text{th}}$ particle.
Recall the ideal gas law equation
$$pV=nRT$$
Lets say we have $m$ components in the gas mixture, with number of moles of each being $n_1$,$n_2$,$n_3$,...,$n_m$. Let $n$ denote sum of all moles. Let partial pressure of each be $p_1$,$p_2$,$p_3$,...,$p_m$, and $p$ denote the total pressure.
From the gas law, we get:
$$pV=nRT$$
$$pV=(n_1+n_2+n_3+...+n_m)RT$$
$$pV=n_1RT+n_2RT+n_3RT+...+n_mRT$$
$$pV=p_1V+p_2V+p_3V+...+p_mV$$
$$p=p_1+p_2+p_3+...+p_m$$
I think that's a satisfactorily enough proof.
