Question

A $5.00-$ mol sample of ${\mathrm{NH}}_3$ gas is kept in a $1.92-\mathrm{L}$ container at $300\mathrm{K}$. If the van der Waals equation is assumed to give the correct answer for the pressure of the gas, calculate the percent error made in using the ideal gas equation to calculate the pressure.

Short answer

The percent error is approximately 33.09%.

Step-by-step solution

Identify and Record Given Values

We are given the number of moles $n=5.00\text{ moles}$, volume $V=1.92\text{ L}$, and temperature $T=300\text{ K}$. Additionally, for ${\text{NH}}_3$, the van der Waals constants are $a=4.17{\text{ L}}^2\cdot {\text{atm/mol}}^2$ and $b=0.0371\text{ L/mol}$.

Calculate Pressure Using Ideal Gas Law

The ideal gas law is $PV=nRT$. First, convert the R constant to the appropriate unit: $R=0.0821\text{ L atm/mol K}$. Solve for pressure $P$: $$P=\frac{nRT}{V}=\frac{(5.00)\times (0.0821)\times (300)}{1.92}\approx 64.11\text{ atm}$$.

Apply the van der Waals Equation

The van der Waals equation is $$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$$. First, calculate the correction terms: $\frac{a{n}^2}{V^2}=\frac{4.17\times (5.00{)}^2}{(1.92{)}^2}\approx 28.34$, and $nb=5.00\times 0.0371=0.1855\text{ L}$. Substitute these values into the van der Waals equation and solve for $P$: $$(P+28.34)(1.92-0.1855)=5.00\times 0.0821\times 300$$. Solve for $P$: $P\approx 48.19\text{ atm}$.

Calculate Percent Error

Percent error is calculated as $\left(\frac{\text{Ideal Pressure}-\text{van der Waals Pressure}}{\text{van der Waals Pressure}}\right)\times 100\mathrm{\%}$. $$\text{Percent Error}=\frac{64.11-48.19}{48.19}\times 100\mathrm{\%}\approx 33.09\mathrm{\%}$$.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the behavior of an ideal gas. It is expressed as:
$$PV=nRT$$
where:

• $P$ represents the pressure of the gas
• $V$ is the volume occupied by the gas
• $n$ is the number of moles of gas
• $R$ is the universal gas constant
• $T$ is the temperature in Kelvin

This equation assumes that the gas particles are point masses with no volume and do not interact except for elastic collisions. It works well under fairly low pressure and high temperature, where intermolecular forces are negligible.
In the example given, we calculated the pressure ($P$) using these known values:$n=5.00$ moles, $V=1.92$ L, and $T=300$ K with the constant $R=0.0821$ L atm/mol K. The calculated pressure for ammonia gas using the ideal gas law was about $64.11$ atm.

Percent Error Calculation

Percent error is an important concept when evaluating the accuracy of experimental or theoretical results. It helps us understand how much a calculated value deviates from an accepted or true value. The formula to calculate percent error is as follows:
$$\text{Percent Error}=\left(\frac{\text{Experimental Value}-\text{Theoretical Value}}{\text{Theoretical Value}}\right)\times 100\mathrm{\%}$$In this context, the theoretical value is the pressure obtained using the van der Waals equation, which corrects for non-ideal gas behavior. The experimental value is the pressure calculated using the ideal gas law.
For ammonia gas, the ideal gas pressure is $64.11$ atm, and the van der Waals pressure is $48.19$ atm. The percent error was calculated as approximately $33.09\mathrm{\%}$. This indicates a significant deviation due to non-ideal behavior of ammonia under the given conditions.

Pressure Calculation

Calculating pressure accurately is crucial when dealing with gases, especially under conditions where deviations from ideal behavior occur. Two common equations are used for this purpose: the Ideal Gas Law and the van der Waals equation.
- **Ideal Gas Law:** - Assumes ideal behavior with no interactions between gas particles. - Useful for initial approximations.- **Van der Waals Equation:** - Accounts for intermolecular forces and the volume occupied by gas particles through van der Waals constants $a$ and $b$. - Expressed as: $$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$$In the exercise, the ideal gas law gave a pressure of $64.11$ atm, while the van der Waals equation provided a more accurate pressure of $48.19$ atm. The differences highlighted the non-ideal behavior of gases like ammonia at high pressures and low temperatures.

Ammonia Gas

Ammonia, denoted as ${\text{NH}}_3$, is a commonly studied gas in chemistry due to its unique properties and wide range of applications.
Properties of ammonia include:

• A polar molecule due to its trigonal pyramidal shape and significant electronegativity difference between nitrogen and hydrogen.
• Higher boiling point compared to non-polar gases of similar molecular weight because of its ability to form hydrogen bonds.
• Often exhibits non-ideal behavior, especially noticeable under varying temperatures and pressures, necessitating the use of equations like van der Waals for accurate predictions.

In the given problem, ammonia's behavior was modeled using both the ideal gas law and the van der Waals equation to reflect its real-world deviations from ideality. Understanding ammonia's properties helps explain why calculations using simple ideal gas assumptions may lead to significant percent errors.