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Chapter 5: Problem 40

Ideal gas particles are assumed to be volume less and to neither attract nor repel each other. Why are these assumptions crucial to the validity of Dalton鈥檚 law of partial pressures?

Short Answer

Expert verified
The assumptions of ideal gas particles being volumeless and having no interactions are crucial to the validity of Dalton's Law of Partial Pressures because they simplify the behavior of gas particles, allowing scientists to predict and calculate gas properties. By assuming gas particles have no volume and interactions, we can treat each particle independently and simply add their partial pressures together, without any complications due to their size or interaction, to get the total pressure as described by Dalton's Law: \(P_{total} = P_1 + P_2 + \cdots + P_n\). Without these assumptions, accurately predicting the total pressures of mixed gases using Dalton's Law would be much more difficult.

Step by step solution

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1. Understanding Ideal Gas Assumptions

The Ideal Gas Law (PV = nRT) is based on two key assumptions: (a) gas particles have no volume compared to the space between them, and (b) gas particles have no interactions with each other (neither attractive nor repulsive forces). These assumptions simplify the behavior of gas particles, allowing scientists to predict and calculate gas properties.
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2. Dalton's Law of Partial Pressures

Dalton's Law states that in a mixture of ideal gases, the total pressure is equal to the sum of the partial pressures of the individual gas components. Mathematically, we can write it as: \(P_{total} = P_1 + P_2 + \cdots + P_n\) Where \(P_{total}\) is the total pressure, and \(P_1, P_2, \cdots, P_n\) are the partial pressures of the individual gas components.
03

3. Volumeless Gas Particles

The assumption of volumeless gas particles is important for the validity of Dalton's Law because it allows gas particles to be considered independently without worrying about their size. As the particles are assumed to be volumeless, their contribution to the total pressure is dependent only on the forces they exert on the container walls and not on the sizes of particles. This means each particle can be treated independently, and their partial pressures can be added up to get the total pressure.
04

4. No Interactions Between Gas Particles

The assumption of no interaction between gas particles is also crucial to the validity of Dalton's Law because it prevents any interference between the particles that could affect their contribution to the total pressure. If gas particles attracted or repelled each other, their pressures wouldn't be independent, and they couldn't be added together as simply as in Dalton's Law.
05

5. Conclusion

The assumptions of ideal gas particles being volumeless and having no interactions are crucial to the validity of Dalton's Law of Partial Pressures because they allow the pressure of individual gas particles to be added together without any complication due to their size or interaction. If the assumptions didn't hold, it would be much more difficult to accurately predict the total pressures of mixed gases using Dalton's Law.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law Assumptions
The Ideal Gas Law is based on certain assumptions that simplify the study and understanding of gases. These assumptions include:
  • Gas particles are assumed to have no volume. This means their size is negligible compared to the space they occupy.
  • Gas particles are assumed to exert no forces on each other, meaning there are no attractions or repulsions between them.
These assumptions are necessary for the law to function predictably. They allow scientists to create models that successfully predict how gases behave under different conditions. In real-world applications, these assumptions help in simplifying calculations and understanding gas mixtures, like those found in air.
Ideal Gases
Ideal gases are hypothetical gases that perfectly adhere to the gas laws, including the Ideal Gas Law (PV = nRT). Their behavior is predictable because they follow specific assumptions:
  • They move with constant velocity.
  • They don鈥檛 interact with each other.
  • Their collisions are perfectly elastic, meaning no energy is lost.
In reality, no gas is truly ideal. However, most gases behave as ideal gases under many conditions, such as at high temperatures and low pressures. This ideal behavior allows scientists and chemists to use simple calculations to predict and understand various gas-related phenomena.
Partial Pressures
Partial pressures refer to the pressure exerted by an individual gas within a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of all the individual gases in the mixture.
This can be mathematically expressed as:
\[ P_{total} = P_1 + P_2 + ext{...} + P_n \]
Each gas in a mixture acts independently when applying Dalton鈥檚 Law, assuming ideal gas behavior. The pressure each gas contributes is directly related to its proportion in the mixture. Without considering interactions or volume, the task of calculating total pressure becomes straightforward by adding up each partial pressure.
Gas Particle Interactions
When considering gas particle interactions, it's important to know that ideal gases assume no interactions between particles. This means:
  • No attractive forces: Particles do not pull towards each other.
  • No repulsive forces: Particles do not push away from each other.
These assumptions ensure that each gas in a mixture contributes independently to the total pressure. If gas particles interacted, either by attraction or repulsion, their behavior and, subsequently, the pressure they exert would change. This would violate Dalton鈥檚 Law, as the partial pressures would not be truly additive. Understanding that no interactions occur simplifies the estimation of pressures in complex gas mixtures.

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Most popular questions from this chapter

Which of the following statements is(are) true? a. If the number of moles of a gas is doubled, the volume will double, assuming the pressure and temperature of the gas remain constant b. If the temperature of a gas increases from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C},\) the volume of the gas would double, assuming that the pressure and the number of moles of gas remain constant. c. The device that measures atmospheric pressure is called a barometer. d. If the volume of a gas decreases by one half, then the pressure would double, assuming that the number of moles and the temperature of the gas remain constant.

You have a helium balloon at 1.00 atm and \(25^{\circ} \mathrm{C} .\) You want to make a hot-air balloon with the same volume and same lift as the helium balloon. Assume air is 79.0\(\%\) nitrogen and 21.0\(\%\) oxygen by volume. The 鈥渓ift鈥� of a balloon is given by the difference between the mass of air displaced by the balloon and the mass of gas inside the balloon. a. Will the temperature in the hot-air balloon have to be higher or lower than \(25^{\circ} \mathrm{C} ?\) Explain. b. Calculate the temperature of the air required for the hot-air balloon to provide the same lift as the helium balloon at 1.00 atm and \(25^{\circ} \mathrm{C}\) . Assume atmospheric conditions are 1.00 atm and \(25^{\circ} \mathrm{C} .\)

One of the chemical controversies of the nineteenth century concerned the element beryllium (Be). Berzelius originally claimed that beryllium was a trivalent element (forming \(\mathrm{Be}^{3+}\) ions) and that it gave an oxide with the formula \(\mathrm{Be}_{2} \mathrm{O}_{3}\) . This resulted in a calculated atomic mass of 13.5 for beryllium. In formulating his periodic table, Mendeleev proposed that beryllium was divalent (forming \(\mathrm{Be}^{2+}\) ions) and that it gave an oxide with the formula BeO. This assumption gives an atomic mass of \(9.0 .\) In \(1894,\) A. Combes (Comptes Rendus \(1894,\) p. 1221 ) reacted beryllium with the anion \(C_{5} \mathrm{H}_{7} \mathrm{O}_{2}^{-}\) and measured the density of the gaseous product. Combes's data for two different experiments are as follows: $$\begin{array}{lll}{\text { Mass }} & {0.2022 \mathrm{g}} & {0.2224 \mathrm{g}} \\ {\text { Volume }} & {22.6 \mathrm{cm}^{3}} & {26.0 \mathrm{cm}^{3}} \\ {\text { Temperature }} & {13^{\circ} \mathrm{C}} & {17^{\circ} \mathrm{C}} \\ {\text { Pressure }} & {765.2 \mathrm{mm} \mathrm{Hg}} & {764.6 \mathrm{mm}}\end{array}$$ If beryllium is a divalent metal, the molecular formula of the product will be \(\mathrm{Be}\left(\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{O}_{2}\right)_{2} ;\) if it is trivalent, the formula will be \(\mathrm{Be}\left(\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{O}_{2}\right)_{3} .\) Show how Combes's data help to confirm that beryllium is a divalent metal.

Consider the following apparatus: a test tube covered with a non permeable elastic membrane inside a container that is closed with a cork. A syringe goes through the cork. a. As you push down on the syringe, how does the membrane covering the test tube change? b. You stop pushing the syringe but continue to hold it down. In a few seconds, what happens to the membrane?

Consider the following samples of gases at the same temperature. Arrange each of these samples in order from lowest to highest: a. pressure b. average kinetic energy c. density d. root mean square velocity Note: Some samples of gases may have equal values for these attributes. Assume the larger containers have a volume twice the volume of the smaller containers, and assume the mass of an argon atom is twice the mass of a neon atom.
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