Question

In a \(1 \mathrm{~mm}^{3}\) volume, \(10^{15}\) atoms of argon are at a temperature of \(T=300 \mathrm{~K}\). Calculate the pressure of the gas.

Short answer

The pressure of the gas is approximately \(4.14 \text{ Pa}\).

Step-by-step solution

Understanding the given and required quantities

We are given a volume of \(1 \text{ mm}^3\), which can be converted to \(10^{-6} \text{ m}^3\). The number of argon atoms present is \(10^{15}\) and the temperature is \(300 \text{ K}\). We need to find the pressure of the gas. By using the ideal gas law \(PV = nRT\), we can find the pressure.

Convert atoms to moles

To use the ideal gas law, we need the number of moles instead of the number of atoms. The conversion is done using Avogadro's number \(6.022 \times 10^{23} \text{ atoms/mol}\):\[n = \frac{10^{15}}{6.022 \times 10^{23}}\approx 1.66 \times 10^{-9} \text{ moles}\]

Apply the Ideal Gas Law

The ideal gas law is \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant \(8.314 \text{ J/mol K}\), and \(T\) is temperature. Substituting the known values:\[P \cdot 10^{-6} = 1.66 \times 10^{-9} \times 8.314 \times 300\]

Calculate the Pressure

Solve the equation from the previous step for \(P\):\[P = \frac{1.66 \times 10^{-9} \times 8.314 \times 300}{10^{-6}} = 4.14 \text{ Pa}\]

Key concepts

Pressure Calculation

Calculating the pressure of a gas can seem challenging at first, but with the ideal gas law, it's straightforward. The ideal gas law is an equation that relates the pressure (\(P\)), volume (\(V\)), number of moles (\(n\)), and temperature (\(T\)) of a gas by the formula \(PV = nRT\). This formula is used for gases assumed to behave ideally, meaning they follow certain conditions like moving randomly without interacting extensively with each other.
To find the pressure of the gas, rearrange the formula: \(P = \frac{nRT}{V}\). By inputting the values for the number of moles (\(n\)), the gas constant (\(R = 8.314\, \text{J/mol K}\)), temperature (\(T = 300\,\text{K}\)), and volume (\(V = 10^{-6}\,\text{m}^3\)), you can solve for \(P\), yielding \(P = 4.14\,\text{Pa}\).
The units and the value of the gas constant must match those used for pressure and volume to maintain consistency.

Moles Conversion

For many chemical calculations, the number of particles is more conveniently expressed in terms of moles rather than atoms. This is because a mole is a standard scientific unit that describes a set quantity of particles. Using Avogadro's number, which is \(6.022 \times 10^{23}\,\text{atoms/mol}\), one can easily convert the number of atoms into moles.
Assuming you know the number of argon atoms, which is \(10^{15}\), you can find the number of moles by dividing the number of atoms by Avogadro's number: \(n = \frac{10^{15}\,\text{atoms}}{6.022 \times 10^{23}\,\text{atoms/mol}}\). This works out to approximately \(1.66 \times 10^{-9}\,\text{moles}\).
This conversion is crucial because the ideal gas law requires the quantity of gas to be in moles for accurate calculations.

Avogadro's Number

Avogadro’s number is a fundamental constant in chemistry, allowing conversion between atoms and moles. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole, and it equals \(6.022 \times 10^{23}\).
This large number is essential for calculating how many particles are in a given amount of substance. Since atoms are exceedingly tiny, Avogadro’s number allows scientists to deal with manageable units (moles) instead of staggeringly large numbers of atoms.
Understanding Avogadro's number helps in many areas of chemistry, including determining the amount of reactants and products in a chemical reaction, and it is foundational for working with the ideal gas law, as seen in our pressure calculation.