0.14 Ideal gas law  (Page 7/7)

Observation 3: partial pressures

We referred briefly above to the pressure of mixtures of gases, noting in our measurements leading toBoyle's Law that the total pressure of the mixture depends only on the number of moles of gas, regardless of the types andamounts of gases in the mixture. The Ideal Gas Law reveals that the pressure exerted by a mole of molecules does not depend on whatthose molecules are, and our earlier observation about gas mixtures is consistent with that conclusion.

We now examine the actual process of mixing two gases together and measuring the total pressure. Consider acontainer of fixed volume 25.0L. We inject into that container 0.78 moles ofN ₂ gas at 298 K. From the Ideal Gas Law, we can easily calculate the measured pressure of the nitrogen gas to be 0.763 atm. We now takean identical container of fixed volume 25.0L, and we inject into that container 0.22 moles ofO ₂ gas at 298 K. The measured pressure of the oxygen gas is 0.215 atm. As a third measurement, we inject 0.22 moles ofO ₂ gas at 298 K into the first container which already has 0.78 moles of N ₂ . (Note that the mixture of gases we have prepared is very similar tothat of air.) The measured pressure in this container is now found to be 0.975 atm.

We note now that the total pressure of the mixture of N ₂ and O ₂ in the container is equal to the sum of the pressures of the N ₂ and O ₂ samples taken separately. We now define the partial pressure of each gas in the mixture to be the pressure of each gas as if it were the only gas present. Our measurementstell us that the partial pressure of N ₂ , $P_{N_2}$ , is 0.763 atm, and the partial pressure ofO ₂ , $P_{O_2}$ , is 0.215 atm.

With this definition, we can now summarize our observation by saying that the total pressure of the mixture ofoxygen and nitrogen is equal to the sum of the partial pressures of the two gases. This is a general result: Dalton's Law of Partial Pressures .

Dalton's law of partial pressures

The total pressure of a mixture of gases is the sum of the partial pressures of the component gases in themixture

Review and discussion questions

1. Sketch a graph with two curves showing Pressure vs. Volume for two different values of the number of molesof gas, with $n_2> n_1$ , both at the same temperature. Explain the comparison of the twocurves.
2. Sketch a graph with two curves showing Pressure vs. 1/Volume for two different values of the number ofmoles of gas, with $n_2> n_1$ , both at the same temperature. Explain the comparison of the twocurves.
3. Sketch a graph with two curves showing Volume vs. Temperature for two different values of the number ofmoles of gas, with $n_2> n_1$ , both at the same pressure. Explain the comparison of the twocurves.
4. Sketch a graph with two curves showing Volume vs Temperature for two different values of the pressure ofthe gas, with $P_2> P_1$ , both for the same number of moles. Explain the comparison of thetwo curves.
5. Explain the significance of the fact that, in the volume-temperature experiments, $\frac{\beta }{\alpha }$ is observed to have the same value, independent of the quantity of gas studied and the type of gas studied. What is the significanceof the quantity $\frac{\beta }{\alpha }$ ? Why is it more significant than either $\beta$ or $\alpha$ ?
6. Amonton's Law says that the pressure of a gas is proportional to the absolute temperature for a fixed quantity of gas in a fixedvolume. Thus, $P=k(N, V)T$ . Demonstrate that Amonton's Law can be derived by combining Boyle's Law and Charles'Law.
7. Using Boyle's Law in your reasoning, demonstrate that the "constant" in Charles' Law, i.e. $k_2(N, P)$ , is inversely proportional to $P$ .
8. Explain how Boyle's Law and Charles' Law may be combined to the general result that, for constant quantity ofgas, $\times (P, V)=kT$ .

9. Using Dalton's Law and the Ideal Gas Law, show that the partial pressure of a component of a gas mixturecan be calculated from

   $P_i=P{X}_i$

   Where $P$ is the total pressure of the gas mixture and $X_i$ is the mole fraction of component $i$ , defined by

   $X_i=\frac{n_i}{n_{\mathrm{total}}}$
10. Dry air is 78.084% nitrogen, 20.946% oxygen, 0.934% argon, and 0.033% carbon dioxide. Determine the molefractions and partial pressures of the components of dry air at standard pressure.

11. Assess the accuracy of the following statement:

    Boyle's Law states that $PV=k_1$ , where $k_1$ is a constant. Charles' Law states that $V=k_{2}T$ , where $k_2$ is a constant. Inserting $V$ from Charles' Law into Boyle's Law results in $P{k}_{2}T=k_1$ . We can rearrange this to read $PT=\frac{k_1}{k_2}=\text{a constant}$ . Therefore, the pressure of a gas is inversely proportional to the temperature of the gas.

    In your assessment, you must determine what information is correct or incorrect, provide the correctinformation where needed, explain whether the reasoning is logical or not, and provide logical reasoning where needed.