Question

A 28.2 L volume of $\mathrm{HCl}(\mathrm{g}),$ measured at $742\mathrm{mmHg}$ and ${25.0}^{\circ }\mathrm{C},$ is dissolved in water. What volume of ${\mathrm{NH}}_3(\mathrm{g}),$ measured at $762\mathrm{mmHg}$ and ${21.0}^{\circ }\mathrm{C},$ must be absorbed by the same solution to neutralize the HCl?

Short answer

The calculated volume of NH3 required to neutralize the HCl is the solution.

Step-by-step solution

Convert HCl Conditions to find Moles

First convert the given pressure of $742\mathrm{mmHg}$ to atmospheres, because the Ideal Gas Law works with pressures in atmospheres. The conversion factor is $1\mathrm{atm}=760\mathrm{mmHg}$. Also, temperatures must be in Kelvin for the Ideal Gas Law. Convert ${25.0}^{\circ }\mathrm{C}$ to Kelvin using the relation $K{=}^{\circ }\mathrm{C}+273.15$. Then, apply the Ideal Gas Law: $PV=nRT$. Rearrange this formula to $n=PV/RT$, where P is the pressure in atmospheres, V is the volume in liters, R is the gas constant in $L\cdot atm/(K\cdot mol)$, and T is the temperature in Kelvin

Convert NH3 Conditions to find Volume

The neutralization reaction shows one mole of NH3 is required to neutralize one mole of HCl. So the amount of moles of NH3 required is the same. Now to find the volume of this quantity of NH3, it is necessary to use the Ideal Gas law again. But first, the conditions for NH3 must be converted. The pressure is given as $762\mathrm{mmHg}$, and should be converted to atmospheres. The temperature is given as ${21.0}^{\circ }\mathrm{C}$, and should be converted to Kelvin. To apply the Ideal Gas Law to get V, rearrange to $V=nRT/P$

Calculate Final Volume

Substitute the number of moles of NH3 obtained from Step 1, and the converted values of P, R and T into the formula. Solve this to get the final volume of NH3 gas.

Key concepts

Gas Stoichiometry

Gas stoichiometry involves using the relationships between reactants and products in a chemical reaction involving gases. It allows us to determine quantities such as moles, volumes, or mass required or produced in the reaction. In our problem, gas stoichiometry is used to find the volume of ${\text{NH}}_3(g)$ needed to neutralize a given volume of $\text{HCl}(g)$. Here's how it works:

• Start by using the Ideal Gas Law, $PV=nRT$, to calculate the number of moles of the first gas, $\text{HCl}(g)$.
• Rearrange the equation to find $n:n=\frac{PV}{RT}$.
• Repeat the process for ${\text{NH}}_3(g)$ by using the stoichiometric relationship from the chemical equation (1:1 mole ratio in this case) to find the volume required.

Understanding this concept helps in applying the Ideal Gas Law effectively, allowing you to explore the quantitative aspects of gases in reactions.

Neutralization Reaction

Neutralization reactions occur when an acid and a base react to form water and a salt. In this context, the hydrochloric acid ($\text{HCl}$) reacts with ammonia (${\text{NH}}_3$) in a simple one-to-one ratio:
- **Reaction**: $$\text{HCl}(g)+{\text{NH}}_3(g)\to {\text{NH}}_4\text{Cl}(s)$$
This means that one mole of hydrochloric acid consumes exactly one mole of ammonia. As a result, calculating the volume of ammonia gas required involves the stoichiometric conversion from moles of $\text{HCl}(g)$ to moles (and thus volume) of ${\text{NH}}_3(g)$. This concept is crucial for understanding how amounts of reactants correspond in reactions, especially when using gases.

Temperature Conversion

Temperature must be in Kelvin when working with the Ideal Gas Law. Kelvin is the absolute temperature scale based on the thermodynamic absolute zero. To convert from Celsius to Kelvin, use the simple equation:
- **Conversion Formula**: $K{=}^{\circ }\text{C}+273.15$
In the exercise, the temperatures need conversion for both $\text{HCl}$ and ${\text{NH}}_3$:

• From ${25.0}^{\circ }C$ to Kelvin results in $298.15K$
• From ${21.0}^{\circ }C$ to Kelvin results in $294.15K$

This step is essential because temperature directly affects gas behavior, influencing pressure and volume calculations in gas stoichiometry challenges.

Pressure Conversion

Pressure conversion is another important step when using the Ideal Gas Law because all pressures need to be in the same units, typically atmospheres. The conversion from mmHg to atmospheres can be performed using the relationship:
- **Conversion Factor**: $1\text{atm}=760\text{mmHg}$
In our case:

• Convert $742\text{mmHg}$ to atmospheres for $\text{HCl}$, resulting in $0.975\text{atm}$
• Convert $762\text{mmHg}$ to atmospheres for ${\text{NH}}_3$, resulting in $1.0026\text{atm}$

Accurate pressure conversion is vital because it ensures the constants in the Ideal Gas Law remain consistent, which is necessary for the precise calculation of moles, volume, or other gas properties. This knowledge is indispensable when working on gas-related stoichiometry problems.