Question

A sample of nitrogen gas in a \(4.5-\mathrm{L}\) container at a temperature of \(27^{\circ} \mathrm{C}\) exerts a pressure of 4.1 atm. Calculate the number of moles of gas in the sample.

Short answer

The nitrogen gas sample contains approximately 0.748 moles of gas.

Step-by-step solution

Convert Temperature from Celsius to Kelvin

First, we need to convert the temperature from Celsius to Kelvin. The formula to convert Celsius to Kelvin is \( K = ^{\circ}C + 273.15 \). Given the temperature is \(27^{\circ} \mathrm{C}\), we can calculate:\[ K = 27 + 273.15 = 300.15 \text{ K} \]

Use the Ideal Gas Law Formula

We use the ideal gas law formula, which is \( PV = nRT \), where:- \( P \) is the pressure in atm- \( V \) is the volume in liters- \( n \) is the number of moles- \( R \) is the ideal gas constant, \( 0.0821 \frac{L \, atm}{mol \, K} \)- \( T \) is the temperature in Kelvin

Re-arrange the Ideal Gas Law to Solve for n

Rearrange the equation to solve for the number of moles \( n \):\[ n = \frac{PV}{RT} \]

Plug in Known Values and Solve for n

Substitute the given values into the rearranged formula:\[ P = 4.1 \, \text{atm}, \quad V = 4.5 \, \text{L}, \quad R = 0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}, \quad T = 300.15 \, \text{K} \]Now calculate:\[ n = \frac{4.1 \times 4.5}{0.0821 \times 300.15} \]\[ n = \frac{18.45}{24.656165} \approx 0.748 \text{ moles} \]

Conclusion

The sample of nitrogen gas contains approximately 0.748 moles of gas.

Key concepts

Gas Pressure

Gas pressure is an essential concept in understanding the behavior of gases. It is the force that the gas exerts on the surfaces of its container. To simplify, imagine gas molecules constantly moving and colliding with the walls of a container. Each collision exerts a small force, and collectively, these forces create the pressure.
Gas pressure is often measured in atmospheres (atm) or pascals (Pa). In this exercise, we used the unit atm.

• 1 atm is roughly equivalent to the air pressure at sea level.
• Pressures can change with different conditions such as volume, temperature, and number of molecules in the container.

The Ideal Gas Law, represented by the equation \( PV = nRT \), helps to understand the relationship among pressure (P), volume (V), and temperature (T) in determining the number of moles (n) of a gas.

Temperature Conversion

Temperature conversion is crucial when working with gas laws, especially the Ideal Gas Law. Different temperature scales are used, but Kelvin is the one required for gas law calculations.
To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. This conversion is necessary as gas behavior predictions require absolute temperature measurements.

• The formula is simple: \( K = ^{\circ}C + 273.15 \)
• Kelvin does not have negative values, which simplifies using equations that involve temperature.

In our specific example, the given temperature was \(27^{\circ} \text{C}\), which we converted to 300.15 K.

Moles Calculation

Calculating the number of moles in a gas sample is an application of the Ideal Gas Law. Moles represent the amount of substance and are central to stoichiometry.
The Ideal Gas Law equation \( PV = nRT \) can be rearranged to calculate moles \( n \):

• Use the rearranged formula: \( n = \frac{PV}{RT} \)
• Substitute in the known values for pressure \( P \), volume \( V \), and temperature \( T \) after conversion to Kelvin.

The solution in the exercise involved using the values of 4.1 atm for pressure, 4.5 L for volume, and 300.15 K for temperature, along with the gas constant \( R = 0.0821 \frac{L \, atm}{mol \, K} \). This gave us 0.748 moles of nitrogen gas.

Nitrogen Gas

Nitrogen gas (\( N_2 \)) is a diatomic molecule constituting about 78% of the Earth's atmosphere, making it the most prevalent gas.

• It is colorless, odorless, and mostly inert at standard conditions, making it stable and nonreactive.
• Nitrogen is crucial in various chemical reactions and is an essential element for living organisms.

In the context of the Ideal Gas Law, nitrogen behaves as an ideal gas under typical conditions, allowing the use of \( PV = nRT \) for calculations like determining moles. Although our exercise focuses on nitrogen, the approach is applicable to other ideal gases, assuming similar conditions.