# Confused with a simple ideal gas model

For an ideal gas of two-component ($$A+B$$) mixture at a temperature $$T$$, which expression is correct?

$$p_T=p_A+p_B \tag{1}$$ or $$\rho_T=\rho_A+\rho_B, \tag{2}$$

where $$p_T$$ and $$\rho_T$$ are the total pressure and total mass density, respectively. They are useful when defining the molar fraction $$x_i$$ and mass fraction $$\omega_i$$ of gas mixture:

$$x_i=\frac{p_i}{p_T}, \quad \omega_i=\frac{\rho_i}{\rho_T} \quad i=A \, \text{or} \, B.$$

If no chemical reaction between the two components, let's assume Eq.(1) is correct firstly. According to the ideal gas law, we have $$p_A=\rho_A \frac{R}{M_A}T$$ and $$p_B=\rho_B \frac{R}{M_B}T$$, where $$M_A$$ and $$M_B$$ are the molar mass of the two species, $$R$$ is the universal gas constant. Similarly, for the gas mixture, we have

$$p_T=\rho_T \frac{R}{M_T} T=\frac{\rho_T R T}{x_A M_A+(1-x_A)M_B},$$

where $$x_A=p_A/p_T$$ is the molar fraction of component $$A$$ and $$M_T$$ is the molar mass of the mixture. Substituting $$p_A$$, $$p_B$$ and $$p_T$$ into Eq.(1) and solve for $$\rho_T$$, we found that

$$\rho_T= (\frac{\rho_A}{M_A}+\frac{\rho_B}{M_B})[x_A M_A+(1-x_A)M_B]. \tag{3}$$

Equation (3) seems to be not consistent with Eq.(2).

I believe Eq.(1) should be correct, which states that the total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture. I am confusing when I read the prestigious textbook Transport Phenomena by Bird, Stewart & Lightfoot. Here is a screenshot of the related page.

I don't know what I am doing wrong in deriving Eq.(3). Can anyone please correct me? Thank you very much!

• Thank you! Please see my update. I understand your explanation but still feel confused with how we defined mass fraction with $\omega_i=\rho_i/\sum_i\rho_i$? Specifically, what is $\rho_i$ exactly? Dec 2 '20 at 6:00
• @user55777 isn’t it just $m_{i}/V$ where $V$ is the total volume of the mixture? Dec 2 '20 at 11:11