Question

Nitrogen gas is a product of the thermal decomposition of ammonium dichromate, (NH_), Cr,O_: $$\left(\mathrm{NH}_{4}\right)_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}(s) \rightarrow \mathrm{N}_{2}(g)+\mathrm{Cr}_{2} \mathrm{O}_{3}(s)+4 \mathrm{H}_{2} \mathrm{O}(g)$$ How many liters of \(\mathrm{N}_{2}\) are produced during the decomposition of 100.0 grams of ammonium dichromate \((M=252.07 \mathrm{g} / \mathrm{mol})\) at \(22^{\circ} \mathrm{C}\) and a pressure of 757 torr?

Short answer

Answer: The volume of nitrogen gas produced during the decomposition of 100 grams of ammonium dichromate is 9.697 L at 22°C and 757 torr.

Step-by-step solution

Convert grams of ammonium dichromate to moles

To convert grams of ammonium dichromate to moles, we will use its molar mass, given as 252.07 g/mol. Moles of ammonium dichromate = (100 g) / (252.07 g/mol) = 0.39687 mol

Use stoichiometry to find moles of nitrogen gas produced

The balanced decomposition equation is: $$(NH_4)_2Cr_2O_7(s) \rightarrow N_2(g) + Cr_2O_3(s) + 4H_2O(g)$$ From the balanced equation, 1 mole of (NH_4)_2Cr_2O_7 decomposes to produce 1 mole of N_2 gas. So, moles of nitrogen gas produced = moles of ammonium dichromate decomposed = 0.39687 mol

Convert moles of nitrogen gas to volume using the ideal gas law

The ideal gas law is given by PV = nRT, where P is pressure, V is volume, n is number of moles, R is the ideal gas constant, and T is temperature in Kelvin. To find the volume, we need to rearrange the ideal gas law to V = nRT/P. Since we are given the pressure in torr, we need to convert it to atm: Pressure in atm = (757 torr) * (1 atm / 760 torr) = 0.99605 atm Temperature in Kelvin = (22°C) + 273.15 = 295.15 K Next, we will use the ideal gas constant, R = 0.0821 L atm / (mol K). Now, we can find the volume of nitrogen gas produced using the ideal gas law: Volume = (n * R * T) / P = (0.39687 mol) * (0.0821 L atm / (mol K)) * (295.15 K) / (0.99605 atm) = 9.697 L So, the volume of nitrogen gas produced during the decomposition of 100 grams of ammonium dichromate is 9.697 L at 22°C and 757 torr.

Key concepts

Stoichiometry

Stoichiometry is a branch of chemistry that helps us understand the quantitative relationships between reactants and products in a chemical reaction. In this context, it is used to determine how much of a product, like nitrogen gas, can be produced from a given amount of reactant, such as ammonium dichromate.

When we look at a balanced chemical equation, stoichiometry tells us about the mole ratio of reactants to products. For example, in the thermal decomposition of ammonium dichromate, represented by the equation: \[ (NH_{4})_{2}Cr_{2}O_{7}(s) \rightarrow N_{2}(g) + Cr_{2}O_{3}(s) + 4H_{2}O(g) \] the equation indicates that one mole of ammonium dichromate produces one mole of nitrogen gas. This simple 1:1 relationship is crucial for calculating how many moles of nitrogen gas are produced.

Using stoichiometry, if we know the number of moles of ammonium dichromate, calculating the moles of nitrogen gas becomes straightforward. For instance, if 100 grams of ammonium dichromate are used, you convert this mass into moles using its molar mass (252.07 g/mol) to find 0.39687 moles of ammonium dichromate. According to stoichiometry, this directly gives us the same number of moles of nitrogen gas.

Molar Mass

Molar mass is an essential concept in chemistry and helps convert between grams and moles, which is fundamental for reacting quantities in a chemical equation. It is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance in moles.

• The molar mass of \(\left(NH_{4}\right)_{2}Cr_{2}O_{7}\) is given as 252.07 g/mol. This number signifies that one mole of this compound weighs 252.07 grams.
• To convert grams of a substance into moles, you divide the mass of the substance by its molar mass.

In our example, we start with 100 grams of ammonium dichromate. To find out how many moles this quantity represents, you perform the following calculation: \[ \text{Moles of } (NH_{4})_{2}Cr_{2}O_{7} = \frac{100.0\, \text{g}}{252.07\, \text{g/mol}} = 0.39687 \text{ mol} \] This conversion is crucial for further calculations involving the number of moles of products formed, as per the stoichiometric ratios outlined in the balanced chemical reaction.

Thermal Decomposition

Thermal decomposition is a chemical process where a substance breaks down into simpler compounds or elements upon heating. This type of reaction is endothermic, meaning it absorbs heat from its surroundings.

In this exercise, we are examining the thermal decomposition of ammonium dichromate: \[ (NH_{4})_{2}Cr_{2}O_{7}(s) \rightarrow N_{2}(g) + Cr_{2}O_{3}(s) + 4H_{2}O(g) \] Upon heating, the solid ammonium dichromate decomposes to produce nitrogen gas, chromium(III) oxide as a solid, and water vapor. This is a perfect demonstration of how energy (heat) is used to break chemical bonds in the reactant, resulting in the formation of new substances.

The practical application of this reaction is seen in pyrotechnics and educational demonstrations, where the vivid green chromium(III) oxide and the gas formation are quite dramatic. Understanding thermal decomposition is crucial because it not only illustrates a fundamental type of reaction but also integrates principles of stoichiometry and molar mass to predict and measure the outcomes of chemical processes.