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Chapter 5: Problem 43

An \(11.2-\mathrm{L}\) sample of gas is determined to contain \(0.50 \mathrm{~mol} \mathrm{~N}_{2}\). At the same temperature and pressure, how many moles of gas would there be in a 20.-L sample?

Short Answer

Expert verified
At the same temperature and pressure, there would be approximately \(0.89 \,\text{mol}\) of gas in the 20-L sample.

Step by step solution

01

Understand the avogadro's law

Avogadro's law says that for an ideal gas, at constant temperature and pressure, the volume of the gas is directly proportional to the number of moles of the gas. Mathematically, it can be represented as: \[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \] Here, \(V_1\) and \(V_2\) are the volumes of the gas, and \(n_1\) and \(n_2\) are the number of moles in the respective volumes at the same temperature and pressure.
02

Find the unknown using given values

Now we have all the values required to solve for the unknown number of moles in the 20-L sample. Substitute the known values in the equation and solve for \(n_2\): \[ \frac{11.2\,\text{L}}{0.50\,\text{mol}} = \frac{20\,\text{L}}{n_2} \]
03

Solve for n2

Now, rearrange the equation and solve for \(n_2\): \[ n_2 = \frac{20\,\text{L} \times 0.50\,\text{mol}}{11.2\,\text{L}} \]
04

Calculate the number of moles

Perform the calculation to find the number of moles in the 20-L sample: \[ n_2 = \frac{10}{11.2} \] \[ n_2 \approx 0.89\,\text{mol} \] So, there would be approximately 0.89 moles of gas in the 20-L sample at the same temperature and pressure.

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Key Concepts

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

Ideal Gas
An ideal gas is a theoretical concept used in the realm of chemistry and physics to simplify the study of gases. By definition, an ideal gas perfectly follows certain assumptions.
  • Molecules do not interact with each other (no attraction or repulsion).
  • Molecules move randomly, and their volume is negligible compared to the container's volume.
  • Collisions between molecules are perfectly elastic.
Real gases often deviate from this behavior, but under many conditions, such as at low pressure and high temperature, they behave like an ideal gas. This assumption allows us to employ simpler mathematical models and formulas to predict the behavior of gases. Avogadro's law, which is essential to solving the problem above, relies heavily on these ideal gas assumptions, providing a direct relationship between the volume and quantity of gas.
Volume and Moles Relationship
The relationship between volume and moles in a gas is critical for understanding how gases behave under different conditions. According to Avogadro's Law, when the temperature and pressure remain constant, the volume occupied by a gas is directly proportional to the number of moles of the gas.In simple terms, if you double the number of moles of gas, its volume will also double, assuming nothing else changes. Mathematically, Avogadro's Law can be expressed as:\[ \frac{V_1}{n_1} = \frac{V_2}{n_2} \]This equation can be used to find unknown values, such as the number of moles given a certain volume, by rearranging it appropriately. In the solved exercise, we used this formula to calculate the moles of nitrogen gas in a different volume, given that temperature and pressure remained constant.
Gas Laws
Gas laws are a set of fundamental principles that describe the behavior of gases. They provide mathematical relationships that help predict how gases will respond to changes in pressure, temperature, and volume. Some of the key gas laws include:
  • Boyle's Law: Relates pressure to volume.
  • Charles's Law: Relates volume to temperature.
  • Avogadro's Law: Relates volume to the number of moles.
  • Ideal Gas Law: Combines all of these laws into one equation, \( PV = nRT \).
Each of these laws helps us understand the complex interactions of gas particles in different scenarios. For example, Avogadro's Law, used in the earlier exercise, allows us to examine how the volume of a gas changes as the number of molecules increases or decreases, provided the temperature and pressure stay consistent. Understanding and applying these gas laws is essential for accurately handling problems involving gases in both theoretical and practical settings.

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Most popular questions from this chapter

Sulfur trioxide, \(\mathrm{SO}_{3}\), is produced in enormous quantities each year for use in the synthesis of sulfuric acid. $$ \begin{aligned} \mathrm{S}(s)+\mathrm{O}_{2}(g) & \longrightarrow \mathrm{SO}_{2}(g) \\ 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{SO}_{3}(g) \end{aligned} $$ What volume of \(\mathrm{O}_{2}(g)\) at \(350 .{ }^{\circ} \mathrm{C}\) and a pressure of \(5.25 \mathrm{~atm}\) is needed to completely convert \(5.00 \mathrm{~g}\) sulfur to sulfur trioxide?

Small quantities of hydrogen gas can be prepared in the laboratory by the addition of aqueous hydrochloric acid to metallic zinc. $$ \mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ Typically, the hydrogen gas is bubbled through water for collection and becomes saturated with water vapor. Suppose \(240 . \mathrm{mL}\) of hydrogen gas is collected at \(30 .{ }^{\circ} \mathrm{C}\) and has a total pressure of \(1.032\) atm by this process. What is the partial pressure of hydrogen gas in the sample? How many grams of zinc must have reacted to produce this quantity of hydrogen? (The vapor pressure of water is 32 torr at \(30^{\circ} \mathrm{C}\).)

The steel reaction vessel of a bomb calorimeter, which has a volume of \(75.0 \mathrm{~mL}\), is charged with oxygen gas to a pressure of 145 atm at \(22^{\circ} \mathrm{C}\). Calculate the moles of oxygen in the reaction vessel.

Do all the molecules in a 1 -mol sample of \(\mathrm{CH}_{4}(g)\) have the same kinetic energy at \(273 \mathrm{~K}\) ? Do all molecules in a \(1-\mathrm{mol}\) sample of \(\mathrm{N}_{2}(g)\) have the same velocity at \(546 \mathrm{~K}\) ? Explain.

You have a sealed, flexible balloon filled with argon gas. The atmospheric pressure is \(1.00 \mathrm{~atm}\) and the temperature is \(25^{\circ} \mathrm{C}\). Assume that air has a mole fraction of nitrogen of \(0.790\), the rest being oxygen. a. Explain why the balloon would float when heated. Make sure to discuss which factors change and which remain constant, and why this matters. Be complete. b. Above what temperature would you heat the balloon so that it would float?
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