Question

As \(\mathrm{NH}_{3}(g)\) is decomposed into nitrogen gas and hydrogen gas at constant pressure and temperature, the volume of the product gases collected is twice the volume of \(\mathrm{NH}_{3}\) reacted. Explain. As \(\mathrm{NH}_{3}(g)\) is decomposed into nitrogen gas and hydrogen gas at constant volume and temperature, the total pressure increases by some factor. Why the increase in pressure and by what factor does the total pressure increase when reactants are completely converted into products? How do the partial pressures of the product gases compare to each other and to the initial pressure of \(\mathrm{NH}_{3} ?\)

Short answer

When NH3 decomposes into N2 and H2 at constant pressure and temperature, the volume of product gases (1 mole N2 + 3 moles H2 = 4 moles) is twice the volume of NH3 reacted due to Avogadro's law. At constant volume and temperature, the total pressure increases by a factor of 2 as the number of moles of gas increases. The partial pressure of N2 is half the initial pressure of NH3, while the partial pressure of H2 is 1.5 times the initial pressure of NH3.

Step-by-step solution

Balanced Chemical Equation

First, let's write down the balanced chemical equation for the decomposition of ammonia gas: \(2 NH_{3}(g) \rightarrow N_{2}(g) + 3 H_{2}(g)\) This equation tells us that two moles of ammonia gas decompose to form one mole of nitrogen gas and three moles of hydrogen gas.

Volume Relationship

Using Avogadro's law, we know that at constant temperature and pressure, equal volumes of gases contain the same number of moles. From the balanced chemical equation, we can see that 2 moles of NH3 form 1 mole of N2 and 3 moles of H2, making a total of 4 moles of product gases. The volume relationship is as follows: \(V_{NH_{3}} \times \frac{2 moles\: NH_{3}}{2 moles\: NH_{3}} = V_{products} \times \frac{4 moles\: products}{2 moles\: NH_{3}}\) Therefore, the volume of product gases is twice the volume of NH3 reacted: \(V_{products} = 2 V_{NH_{3}}\)

Pressure Relationship

During the decomposition of NH3 at constant volume and temperature, the number of moles of gas increases, as mentioned before. According to the ideal gas law \(PV = nRT\), if the volume and temperature are constant, an increase in the number of moles of gas will lead to an increase in pressure. As the decomposition proceeds, the pressure of NH3 decreases, while the pressures of N2 and H2 increase. When the reactants are completely converted into products, the initial pressure of NH3 is distributed among the products. From the balanced chemical equation, we can see that the total moles of product gases (1 mole of N2 + 3 moles of H2) = 4 moles, whereas the reactant side has 2 moles of NH3. The total pressure will increase by a factor of 2 (4 moles of product gases compared to 2 moles of reactant gases).

Partial Pressures

To compare the partial pressures of N2 and H2 with initial pressure of NH3, we can use the mole fraction relationship. Partial pressure of N2: \(\frac{1\: mole\: N2}{4\: moles\: total\: products} \times P_{initial\:NH3} = \frac{1}{2} P_{initial\:NH3}\) Partial pressure of H2: \(\frac{3\: moles\: H2}{4\: moles\: total\: products} \times P_{initial\:NH3} = \frac{3}{2} P_{initial\:NH3}\) The partial pressure of N2 is half the initial pressure of NH3, while the partial pressure of H2 is 1.5 times the initial pressure of NH3.

Key concepts

Avogadro's Law

Avogadro's Law is a fundamental concept in chemistry that offers a simple relationship between the volume and the amount of gas, provided that the pressure and temperature remain constant. It states that equal volumes of all ideal gases, at the same temperature and pressure, contain the same number of molecules. This principle is crucial when analyzing reactions involving gases, such as the decomposition of ammonia (\(NH_{3}(g)\)).

In this particular reaction, two moles of ammonia yield a total of four moles of gas products (one mole of nitrogen and three moles of hydrogen). Avogadro's Law explains why the volume of gases produced is twice the volume of ammonia consumed: since the volume is directly proportional to the number of moles (assuming pressure and temperature are constant), doubling the number of moles results in doubling the volume.

Ideal Gas Law

The ideal gas law is a powerful equation in chemistry and physics that relates the pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and number of moles (\(n\)) of an ideal gas through the equation \(PV = nRT\), where \(R\) is the gas constant. This equation is extremely useful for predicting the behavior of gases under different conditions.

In the context of ammonia decomposition, as the reaction proceeds at constant volume and temperature, the number of moles of gases increases. The ideal gas law tells us that an increase in moles at constant volume and temperature results in an increase in pressure. So, when ammonia gas decomposes into nitrogen and hydrogen, the total pressure of the system increases because there are more gas particles pushing against the walls of the container.

Chemical Equation Balancing

Balancing chemical equations is vital for correctly describing the stoichiometry of a chemical reaction. A balanced equation ensures that the same number of atoms for each element is present on both reactant and product sides of the reaction. This concept is strictly tied to the law of conservation of mass, which states that matter is neither created nor destroyed in a chemical reaction.

For the decomposition of ammonia, the balanced chemical equation is: \(2 NH_{3}(g) \rightarrow N_{2}(g) + 3 H_{2}(g)\). It shows that two moles of ammonia decompose into one mole of nitrogen gas and three moles of hydrogen gas. This balanced equation is essential not only for understanding the chemical process but also for predicting the volume and pressure changes of gases involved as expressed by Avogadro's law and the ideal gas law.

Partial Pressure

Partial pressure is a concept in chemistry that refers to the pressure a single gas in a mixture would exert if it occupied the entire volume of the mixture at the same temperature. The total pressure of a gas mixture is the sum of the partial pressures of each individual gas. Dalton's Law of Partial Pressures can be used to calculate the pressure of each gas based on its mole fraction.

With respect to the ammonia decomposition, the partial pressures of nitrogen and hydrogen can be determined using their respective mole fractions multiplied by the original pressure of ammonia. For example, since nitrogen constitutes one fourth of the mole quantity after complete decomposition, its partial pressure is one half of the original pressure of ammonia, according to the balanced chemical equation. Likewise, hydrogen's partial pressure, being three fourths of the total amount, is three halves of the ammonia's original pressure. Understanding partial pressures is essential for predicting how gas mixtures will behave and interact.