Question

The steel reaction vessel of a bomb calorimeter, which has a volume of 75.0 \(\mathrm{mL}\) , is charged with oxygen gas to a pressure of 14.5 atm at \(22^{\circ} \mathrm{C}\) . Calculate the moles of oxygen in the reaction vessel.

Short answer

The moles of oxygen gas in the reaction vessel can be calculated using the Ideal Gas Law Equation, PV = nRT. Given the volume V = 0.075 L, pressure P = 14.5 atm (converted to 1468712.5 Pa), and temperature T = 295 K, the moles of oxygen (n) can be calculated as: n = (1468712.5 Pa) × (0.075 L) / (8.314 J/molK × 295 K) n ≈ 0.0042 mol Therefore, there are approximately 0.0042 moles of oxygen gas in the reaction vessel.

Step-by-step solution

List the given values and convert all to SI units

Given: Volume, V = 75.0 mL = 0.075 L (1 L = 1000 mL) Pressure, P = 14.5 atm (1 atm = 101325 Pa, we'll convert later) Temperature, T = 22℃ = 295 K ( K = ℃ + 273)

Convert units

1. Pressure conversion 1 atm = 101325 Pa 14.5 atm = 14.5 × 101325 Pa = 1468712.5 Pa So now, P = 1468712.5 Pa

Rearrange the Ideal Gas Law Equation to solve for moles (n)

The Ideal Gas Law Equation is given by: PV = nRT We have to calculate the moles of oxygen gas, so we'll rearrange the equation to solve for n: n = PV / RT

Substitute the values and calculate n

Now, we can substitute the values of P, V, R, and T into the equation: n = (1468712.5 Pa) × (0.075 L) / (8.314 J/molK × 295 K) n ≈ 0.0042 mol So, there are approximately 0.0042 moles of oxygen gas in the reaction vessel.

Key concepts

Bomb Calorimeter

A bomb calorimeter is a sophisticated device used to measure the heat of combustion of a sample. It primarily helps in understanding the energy changes during a chemical reaction. The device consists of a robust, sealed container where the reaction occurs, primarily designed to withstand high pressures. Typically, it includes a metal vessel submerged in a water bath to absorb the heat released by the reaction.

The design ensures that the heat released is contained, providing accurate measurement of the total energy change. The bomb calorimeter is very effective in experiments involving gases like oxygen, as seen in this example.

• This method ensures that the reaction occurs at a constant volume, meaning no work is done by the system except for potential pressure changes.
• The oxygen gas, used to combust the sample, is measured and controlled within the chamber accurately.

The results give valuable data for calculating enthalpies of various reactions, which is important for understanding reaction energetics.

Moles Calculation

Calculating the moles of a substance is an integral part of many chemistry exercises, including this one involving the ideal gas law. The ideal gas law equation is given by: \[ PV = nRT \]Here's a breakdown of what each symbol represents:

• \(P\) stands for pressure
• \(V\) is the volume
• \(n\) denotes the number of moles
• \(R\) is the ideal gas constant, \(8.314 \, \text{J/mol K}\)
• \(T\) is for temperature in Kelvin

When calculating moles, rearrange the formula to solve for \(n\): \[ n = \frac{PV}{RT} \]
This equation helps to find the number of moles of a gas when provided with the other variables' values.

It's crucial to ensure all units are correct before substitution. In this example, pressure is converted from atm to Pascal, and temperature is in Kelvin. The correct application of this formula yields precisely how much substance is present, crucial for reactions in a bomb calorimeter.

Unit Conversion

In scientific calculations, proper unit conversion is crucial for accuracy. It ensures consistency and compatibility across different units of measurement. In the context of this problem, unit conversion is essential for applying the ideal gas law correctly.

For volume, converting milliliters to liters is necessary. The conversion is straightforward, with 1 liter equal to 1000 milliliters. Thus, 75 mL is equivalent to 0.075 L.

Pressure is another key element requiring conversion from atmospheres (atm) to Pascals (Pa), as the ideal gas constant \(R\) is usually expressed with Pascals. The conversion uses the factor that 1 atm equals 101325 Pa. Consequently, 14.5 atm is converted to 1468712.5 Pa.

Furthermore, for temperature, converting from Celsius to Kelvin is necessary. The conversion is simple: \[ K = °C + 273 \]
Thus, 22°C converts to 295 K.

These conversions are essential to apply \[ PV = nRT \] correctly and to get the accurate number of moles for the reaction in our bomb calorimeter.