Chemistry - Ideal Gas Question with Solution | TestHub

Dalton's Law: Suppose a mixture of two ideal gases, A and B, is contained in a volume V at a temperature T.

Then, since each gas is ideal, we can write:

$$P_A=\frac{n_{A}RT}{V}$$

$$P_B=\frac{n_{B}RT}{V}$$

That is, in the mixture, each gas exerts a pressure that is the same as it would exert if it were present alone, and this pressure is proportional to the number of moles of the gas present. The quantities $P_A$ and $P_B$ are called the partial pressures of A and B, respectively. According to Dalton's law of partial pressures, the total pressure, $P_t$, exerted on the walls of the vessel is the sum of the partial pressures of the two gases:

$$P_t=P_A+P_B=(n_A+n_B)\frac{RT}{V}$$

The expression can be generalized to apply to a mixture of any number of gases. The result is:

$$P_t=\sum _i{P}_i=\left(\sum _i{n}_i\right)\frac{RT}{V}\dots \left(i\right)$$

where 'i' is an index that identifies each component in the mixture, and the symbol ${\sum }_i$ stands for the operation of adding all the indexed quantities together. Another useful expression of the law of partial pressures is obtained by writing:

$$P_A=n_A\frac{RT}{V},P_t=\frac{RT}{V}\sum _i{n}_i,\frac{P_A}{P_t}=\frac{n_A}{{\sum }_i{n}_i}$$

$$P_A=P_t\left(\frac{n}{{\sum }_i{n}_i}\right)\dots \left(ii\right)$$

The quantity $\frac{n_A}{{\sum }_i{n}_i}$ is called the mole fraction of component A, and the equation says that the partial pressure of any component, such as component A, is the total pressure of the mixture multiplied by $\frac{n_A}{{\sum }_i{n}_i}$the fraction of the total moles which are component A.