Problem 22 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Car safoty airbags inflate when the car undergoes a sudden deceleration, setting oft a reaction which produces a large amount of gas. One of the reactions used is \(\mathrm{NaN}_{3}(\mathrm{s}) \rightarrow \mathrm{Na}(\mathrm{s})+1.5 \mathrm{N}_{2}(\mathrm{g}) \quad \Delta, \mathrm{H}^{0}_{29 \mathrm{e}}=-21.7 \mathrm{kJmol}^{-1}\) (Section \(13.3)\) (a) What is the value of \(\Delta_{1} H_{294}\) for \(\mathrm{NaN}_{3} ?\) (b) \(\quad\) An airbag system is inflated by 2.4 mol (around \(60 \mathrm{dm}^{3}\) ) of \(\mathrm{N}_{2}(\mathrm{q})\). Calculate the enthalpy change when this amount of nitrogen is produced at \(298 \mathrm{K}\) (c) Assuming this energy all goes into heating the \(\mathrm{N}_{2}(\mathrm{g}),\) what Will be the final ternperature of the gas? \(\left(C_{p} \text { for } N_{2}(g)=\right.\) \(\left.29.1 \mathrm{JK}^{-1} \mathrm{mol}^{-1} .\right)\)

Step-by-Step Solution

Answer

(a) \(-21.7 \text{ kJ/mol}\); (b) \(-34.72 \text{ kJ}\); (c) 795.3 K.

Step 1: Understanding the Reaction Enthalpy

The reaction given is: \( \mathrm{NaN}_{3}(\mathrm{s}) \rightarrow \mathrm{Na}(\mathrm{s}) + 1.5 \mathrm{N}_{2}(\mathrm{g}) \). The enthalpy change \( \Delta H^0 \) for this reaction is \(-21.7 \text{ kJ/mol} \). This means 21.7 kJ is released when 1 mole of \( \mathrm{NaN}_3 \) decomposes into \( \mathrm{Na} \) and \( 1.5 \mathrm{N}_2 \).

Step 2: Calculate Enthalpy Change for Producing 2.4 mol of N2

We know that 1 mole of \( \mathrm{NaN}_3 \) produces \(1.5\) moles of \( \mathrm{N}_2 \). Thus, to produce 2.4 moles of \( \mathrm{N}_2 \), we need: \[ \text{Moles of } \mathrm{NaN}_3 = \frac{2.4}{1.5} = 1.6 \text{ moles} \] Since \( \Delta H = -21.7 \text{ kJ/mol} \) for 1 mole of \( \mathrm{NaN}_3 \), the enthalpy change for 1.6 moles of \( \mathrm{NaN}_3 \) is: \[ \Delta H = 1.6 \times (-21.7 \text{ kJ/mol}) = -34.72 \text{ kJ} \]

Step 3: Calculate Final Temperature of N2 Gas

The energy released \(-34.72 \text{ kJ}\) will be absorbed by the produced \( \mathrm{N}_2 \) gas. We use the formula \( q = mC_p\Delta T \) to find the final temperature where \( q = -34.72 \text{ kJ} = -34720 \text{ J} \), \( C_p = 29.1 \text{ JK}^{-1} \text{mol}^{-1} \), and \( m = 2.4 \text{ mol} \).\[ \Delta T = \frac{34720}{2.4 \times 29.1} = \frac{34720}{69.84} \approx 497.3 \text{ K} \] Assuming initial temperature is \(298K\), the final temperature is: \[ 298 + 497.3 \approx 795.3 \text{ K} \]

Key Concepts

Chemical Reactions

In the fascinating world of chemistry, a chemical reaction occurs when substances, known as reactants, undergo a transformation to create new substances, termed products. This process involves the breaking and forming of bonds, resulting in substances with different properties than what you started with. In our exercise featuring the decomposition of sodium azide (NaN₃), this transformation is dramatic. When NaN₃ breaks down, it produces sodium metal and gaseous nitrogen (N₂).
This is particularly interesting because in the context of airbags in vehicles, this reaction is executed rapidly to create a large volume of nitrogen gas. This gas fills the airbag in a mere fraction of a second, providing a cushion that reduces the risk of injury in accidents.

• Reactants: These are the starting substances in a chemical reaction; in our case, NaN₃.
• Products: These are the end substances after the chemical reaction; here, it's Na and N₂.
• Decomposition Reaction: This is a type of chemical reaction where one compound breaks down into two or more components.

By comprehending the nature of chemical reactions, such as this decomposition reaction, we recognize how they govern the rapid inflation of an airbag thus enhancing passenger safety.

Thermodynamics

Thermodynamics plays a critical role in understanding and predicting the energy changes associated with chemical reactions. In any chemical reaction, energy can either be absorbed or released, a concept which is encapsulated by the term 'enthalpy change' (ΔH). Enthalpy is a measure of total energy in the system, heavily influenced by temperature, pressure, and the state of the substances involved.
In our example with NaN₃ decomposition, the reaction releases energy, as indicated by a negative value of enthalpy change, ΔH = -21.7 kJ/mol. This exothermic reaction means energy is released as heat, warming the surroundings, which in this case is utilized to fill an airbag promptly. Let's explore key thermodynamic concepts:

• Exothermic Reaction: A reaction that releases energy to the surroundings; the decomposition of NaN₃ is exothermic.
• ΔH (Enthalpy Change): Indicates whether energy is absorbed or released; negative ΔH implies energy release.
• System and Surroundings: In thermodynamics, the system is the part under study (here, NaN₃), and the surroundings encompass everything else.

Understanding these thermodynamic principles sheds light on how energy changes drive applications like airbags, ensuring they operate effectively when they are most needed.

Gas Laws

Gas laws describe how gases behave in response to changes in pressure, volume, and temperature. These are crucial for applications that involve the quick production of gases like the inflation of airbags, where controlling these variables is vital. The immediate inflation of the airbag relies fundamentally on the generation of nitrogen gas, governed by these gas laws. The scenario described in the exercise demonstrates practical applications of concepts such as the Ideal Gas Law, which relates the pressure (P), volume (V), and temperature (T) of a gas with the quantity in moles (n) through the equation: \[ PV = nRT \]where \( R \) is the Ideal Gas constant. Let’s examine these vital gas law concepts:

• Volume: The amount of space occupied by the gas, increases rapidly as the reaction occurs.
• Pressure: The force per unit area exerted by the gas, which must be controlled to inflate the airbag without bursting.
• Temperature: A rise in temperature can increase the gas volume; thus, it’s managed carefully.

By understanding these laws, one appreciates the precise balance needed to ensure airbags inflate quickly yet safely. These principles not only explain daily technologies but also underline the impressive capabilities of chemistry and physics in protecting lives.