Problem 32 - General Chemistry: Principles and Modern Applications

Question

What is the pressure, in pascals, exerted by \(1242 \mathrm{g}\) CO(g) when confined at \(-25^{\circ} \mathrm{C}\) to a cylindrical tank \(25.0 \mathrm{cm}\) in diameter and \(1.75 \mathrm{m}\) high?

Step-by-Step Solution

Answer

The pressure exerted by the CO(g) in the tank is 115065 Pa or 115 kPa.

Step 1: Calculate the number of moles

The molar mass of CO(g) is \(28.01 g/mol\). Using this, the number of moles (\(n\)) can be calculated using the formula \(n = \frac{mass}{molar \, mass}\), so \(n = \frac{1242g}{28.01 g/mol} = 44.34 mol\).

Step 2: Calculate the volume of the tank

The volume of a cylinder is given by \(V= \pi r^2h\), where \(r\) is the radius, and \(h\) is the height. As the diameter is given, the radius \(r = \frac{d}{2} = \frac{25 cm}{2}\). Convert the radius from cm to m as \(r = 0.125 m\) because the volume must be in cubic meters to match the units of the gas constant R. The volume becomes \(V= \pi (0.125 m)^2 \times 1.75 m = 0.086 m^3\).

Step 3: Convert temperature to Kelvin

The temperature is given in Celsius but must be in Kelvin for the Ideal Gas Law, which can be done using the formula \(T(K) = T(°C) + 273.15\). So the temperature is \(-25 °C + 273.15 = 248.15 K\).

Step 4: Use the Ideal Gas Law

The Ideal Gas Law is \(P = \frac{nRT}{V}\), where P is pressure, R is the Ideal gas constant (R = \(8.314 \, J/(mol \cdot K)\)), n is the number of moles, T is the temperature in Kelvin, and V is the volume in cubic meters. Substituting the calculated values in gives \(P = \frac{44.34 mol \times 8.314 J/(mol \cdot K) \times 248.15 K}{0.086 m^3}\)

Step 5: Calculate the pressure

Calculate the pressure by performing the arithmetic to find \(P = 115065 Pa\) or \(115 kPa\). Therefore, the pressure exerted by the CO(g) in the tank is \(115 kPa\) or \(115065 Pa\).

Key Concepts

Moles Calculation

Understanding moles and how to calculate them is key in chemistry, especially when using the Ideal Gas Law. Moles measure the amount of a substance. To find the number of moles from a given mass, you use the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \).

For example, in the exercise, the molar mass of carbon monoxide (CO) is \(28.01 \, g/mol\). Given a mass of \(1242 \, g\), the number of moles \(n\) is calculated by dividing the mass by the molar mass: \( n = \frac{1242 \, g}{28.01 \, g/mol} = 44.34 \, mol\).

This step is crucial because the Ideal Gas Law (\( PV = nRT \)) requires the number of moles to calculate the pressure of a gas. Remember, the molar mass is specific to each element or compound, so always ensure you have the correct values.

Cylinder Volume Calculation

Calculating the volume of the container is essential for determining pressure using the Ideal Gas Law. For a cylindrical tank, volume \(V\) is calculated with the formula: \( V = \pi r^2 h \).

Consider the given exercise: the cylinder has a diameter of \(25.0 \, cm\), which means the radius \(r\) is half of that, or \(12.5 \, cm\). It’s important to convert centimeters to meters for consistent units, so \(r = 0.125 \, m\).

The height \(h\) is given as \(1.75 \, m\). Plug these into the formula to get the volume: \[ V = \pi (0.125 \, m)^2 \times 1.75 \, m = 0.086 \, m^3 \]

Calculating this volume accurately is vital because it directly affects the calculation of pressure in the Ideal Gas Law. This ensures that all units are consistent with the gas constant \(R\), which uses cubic meters.

Temperature Conversion to Kelvin

In the realm of gas calculations, temperature should always be in Kelvin. The Ideal Gas Law, \( PV = nRT \), requires this to maintain consistency with the gas constant \(R\), which operates on the Kelvin scale.

The conversion from Celsius to Kelvin is straightforward: add \(273.15\) to the Celsius temperature. For instance, in the exercise, the temperature is \(-25^{\circ} C\). To convert, simply do \(-25^{\circ} C + 273.15 = 248.15 \, K\).

This step is often overlooked but is essential for accurate calculations within the Ideal Gas framework. Always ensure temperature is expressed in Kelvin to avoid errors in your results.