Question

Dinitrogen monoxide, ${\mathrm{N}}_2\mathrm{O},$ reacts with propane, ${\mathrm{C}}_3{\mathrm{H}}_8,$ to form nitrogen, ${\mathrm{N}}_2$; carbon dioxide, ${\mathrm{CO}}_2$; and water, ${\mathrm{H}}_2\mathrm{O}$. a. Write a balanced chemical equation for this reaction, treating all substances as gases. Include phases in your equation. b. Two reservoirs, separated by a closed valve (of negligible volume), are filled with the reactants (one on each side). The pressure of ${\mathrm{N}}_2\mathrm{O}$ is measured to be 1.4 atm. The pressure of ${\mathrm{C}}_3{\mathrm{H}}_8$ is measured to be 1.2 atm. Each reservoir has a capacity of $1.0\mathrm{L},$ and the temperature of the system is 450 "the partial pressure of ${\mathrm{CO}}_2$ after the reaction is complete? c. What is the partial pressure of ${\mathrm{N}}_2\mathrm{O}$ after the reaction is complete?

Short answer

a. The balanced chemical equation is: $$2N_{2}O(g)+3C_3{H}_8(g)\to 4N_2(g)+9C{O}_2(g)+12H_{2}O(g)$$ b. The partial pressure of CO2 after the reaction is 1.21 atm. c. The partial pressure of N2O after the reaction is 1.14 atm.

Step-by-step solution

a. Write a balanced chemical equation

To balance the chemical equation, we must ensure that there are equal numbers of each type of atom on both sides of the equation. The given reaction is: $$N_{2}O(g)+C_3{H}_8(g)\to N_2(g)+C{O}_2(g)+H_{2}O(g)$$ Balancing the equation: $$2N_{2}O(g)+3C_3{H}_8(g)\to 4N_2(g)+9C{O}_2(g)+12H_{2}O(g)$$ Now, our chemical equation is balanced.

b. Calculate the partial pressure of CO2

We will determine the initial moles of each reactant and use stoichiometry to find the moles of CO2 produced. Then, we'll use the ideal gas law to find the partial pressure of CO2. 1. Calculate the moles of reactants: Use the ideal gas law, PV=nRT, n = PV/RT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/(mol K)), and T is the temperature in Kelvin. For N2O: $n(N_{2}O)=\frac{1.4\text{atm}\times 1.0\text{L}}{(0.0821\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}})\times 450\text{K}}=0.0381\text{mol}$ For C3H8: $n(C_3{H}_8)=\frac{1.2\text{atm}\times 1.0\text{L}}{(0.0821\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}})\times 450\text{K}}=0.0328\text{mol}$ 2. Determine the limiting reactant: From the balanced equation, we see the mole ratio of N2O:C3H8 is 2:3. Divide the moles of each reactant by their respective stoichiometric coefficients: $\frac{n(N_{2}O)}{2}=\frac{0.0381\text{mol}}{2}=0.0191$ $\frac{n(C_3{H}_8)}{3}=\frac{0.0328\text{mol}}{3}=0.0109$ The smallest value is 0.0109, which means C3H8 is the limiting reactant. 3. Calculate the moles of CO2 produced: The mole ratio of C3H8:CO2 is 3:9. $n(C{O}_2)=0.0109\times \frac{9}{3}=0.0328\text{mol}$ 4. Calculate the partial pressure of CO2: Using the ideal gas law, PV=nRT, P = nRT/V: $P(C{O}_2)=\frac{0.0328\text{mol}\times (0.0821\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}})\times 450\text{K}}{1.0\text{L}}=1.21\text{atm}$ The partial pressure of CO2 after the reaction is 1.21 atm.

c. Calculate the partial pressure of N2O

To find the partial pressure of N2O after the reaction, we will calculate the moles of N2O remaining after the reaction and use the ideal gas law. 1. Calculate the moles of N2O remaining: The mole ratio of N2O:C3H8 is 2:3. The moles of N2O consumed = moles of C3H8 consumed × (2/3) = 0.0109 mol × (2/3) = 0.00726 mol Thus, moles of N2O remaining = initial moles of N2O - moles of N2O consumed = 0.0381 mol - 0.00726 mol = 0.0308 mol 2. Calculate the partial pressure of N2O: Using the ideal gas law, PV=nRT, P = nRT/V: $P(N_{2}O)=\frac{0.0308\text{mol}\times (0.0821\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}})\times 450\text{K}}{1.0\text{L}}=1.14\text{atm}$ The partial pressure of N2O after the reaction is 1.14 atm.

Key concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It ensures that chemical reactions obey the conservation of mass, meaning that atoms are neither created nor destroyed. To practice stoichiometry, you'll use the balanced chemical equation, which shows the precise mole ratios between all the reactants and products.

Consider the reaction between dinitrogen monoxide (${\mathrm{N}}_2\mathrm{O}$) and propane (${\mathrm{C}}_3{\mathrm{H}}_8$). The balanced equation for this reaction is:
$$2{\mathrm{N}}_2\mathrm{O}(g)+3{\mathrm{C}}_3{\mathrm{H}}_8(g)\to 4{\mathrm{N}}_2(g)+9{\mathrm{CO}}_2(g)+12{\mathrm{H}}_2\mathrm{O}(g)$$

This equation tells us that 2 moles of ${\mathrm{N}}_2\mathrm{O}$ react with 3 moles of ${\mathrm{C}}_3{\mathrm{H}}_8$ to produce 4 moles of nitrogen gas, 9 moles of carbon dioxide, and 12 moles of water vapor. Using these ratios, you can calculate the amount of product formed from a given amount of reactant, or the amount of one reactant needed to react with a given amount of another reactant. This is known as the mole ratio.

• Start by identifying the number of moles of a given reactant.
• Use the mole ratios from the balanced equation to convert between substances.
• The limiting reactant is the one that will run out first and limit the amount of products formed.

Understanding stoichiometry is crucial for determining products in chemical reactions accurately. It is an essential part of chemical calculations that make real-world reactions predictable.

Ideal Gas Law

The ideal gas law is a powerful equation in chemistry used to relate the pressure, volume, temperature, and number of moles of a gas. This relationship is represented by the formula:$$PV=nRT$$
where:

• $P$ stands for pressure
• $V$ is the volume of the gas
• $n$ is the number of moles
• $R$ is the ideal gas constant (0.0821 L atm/mol K)
• $T$ is the temperature in Kelvin

In the context of reactions involving gases, like the reaction between ${\mathrm{N}}_2\mathrm{O}$ and ${\mathrm{C}}_3{\mathrm{H}}_8$, the ideal gas law allows us to find unknowns such as the number of moles of a gas when the other parameters are known.

For example, to find the number of moles of ${\mathrm{N}}_2\mathrm{O}$ in a reservoir, you would rearrange the equation to solve for $n$:
$$n=\frac{PV}{RT}$$By substituting the known values of pressure, volume, and temperature into the equation, you can calculate the moles of gas present.

This law also helps in finding the partial pressures of gases after reactions by using the same formula to calculate the pressures from the moles of gases left or produced. Understanding the ideal gas law is essential to solving many problems in chemistry involving gases, as it provides a simple model for the behavior of ideal gases.

Balanced Chemical Equations

Balanced chemical equations are essential tools in chemistry, providing a blueprint for the quantitative relationships in a chemical reaction. A balance ensures that the number of each type of atom on the reactant side (left) equals the number on the product side (right). This reflects the law of conservation of mass.

To begin balancing an equation, you write down the unbalanced chemical equation with all reactants and products. For instance, the reaction between ${\mathrm{N}}_2\mathrm{O}$ and ${\mathrm{C}}_3{\mathrm{H}}_8$ is initially:
$${\mathrm{N}}_2\mathrm{O}(g)+{\mathrm{C}}_3{\mathrm{H}}_8(g)\to {\mathrm{N}}_2(g)+{\mathrm{CO}}_2(g)+{\mathrm{H}}_2\mathrm{O}(g)$$Then, you'll adjust the coefficients (numbers in front of the chemical formulas) to balance the atom numbers on each side. The balanced equation for this reaction is:
$$2{\mathrm{N}}_2\mathrm{O}(g)+3{\mathrm{C}}_3{\mathrm{H}}_8(g)\to 4{\mathrm{N}}_2(g)+9{\mathrm{CO}}_2(g)+12{\mathrm{H}}_2\mathrm{O}(g)$$

To balance equations:

• First balance elements that appear only once on each side.
• Balance elements that appear in multiple compounds last, like oxygen.
• Check that all atoms balance and that you have the smallest whole-number coefficients possible.

Balancing chemical equations is a foundational step in performing stoichiometry calculations, as it provides the necessary molar relationships between reactants and products.