Question

Two identical containers hold equal masses of gas, oxygen in one and nitrogen in the other. The gases are held at the same temperature. How does the pressure of the oxygen compare to that of the nitrogen? a) \(p_{\mathrm{O}}>p_{\mathrm{N}}\) b) \(p_{\mathrm{O}}=p_{\mathrm{N}}\) c) \(p_{\mathrm{O}}

Short answer

a) \(p_{\mathrm{O}} > p_{\mathrm{N}}\) b) \(p_{\mathrm{O}} = p_{\mathrm{N}}\) c) \(p_{\mathrm{O}} < p_{\mathrm{N}}\) d) None of the above Answer: c) \(p_{\mathrm{O}} < p_{\mathrm{N}}\)

Step-by-step solution

Write down the Ideal Gas Law equation

The Ideal Gas Law equation is given by: \(PV = nRT\) Where: - P is the pressure of the gas - V is the volume of the container - n is the number of moles of gas - R is the universal gas constant - T is the temperature of the gas

Identify the important variables

Since we are comparing the pressure of the gases, we need to find the ratio of the number of moles for each gas: \(n_{\mathrm{O}}\) = number of moles of oxygen \(n_{\mathrm{N}}\) = number of moles of nitrogen We know that the mass of the gases is equal, and the volume and temperature are also the same.

Calculate the number of moles for each gas

We can calculate the number of moles for each gas using the following formula: \(n = \frac{m}{M}\) Where: - n is the number of moles - m is the mass of the gas - M is the molar mass of the gas For oxygen: - molar mass M = 32 g/mol For nitrogen: - molar mass M = 28 g/mol Given that the masses of the gases are equal (m), we can write the ratio of the number of moles for each gas as follows: \(\frac{n_\mathrm{O}}{n_\mathrm{N}} = \frac{\frac{m}{32}}{\frac{m}{28}} = \frac{28}{32}\)

Use the Ideal Gas Law to compare pressures

We can now use the Ideal Gas Law to compare the pressures of oxygen and nitrogen under the same conditions: \(p_\mathrm{O}V = n_\mathrm{O}RT\) \(p_\mathrm{N}V = n_\mathrm{N}RT\) Since the volumes and temperatures are equal, we can divide both equations: \(\frac{p_\mathrm{O}}{p_\mathrm{N}} = \frac{n_\mathrm{O}}{n_\mathrm{N}} = \frac{28}{32}\)

Determine the relationship between the pressures

From the previous step, we can see that the ratio of the pressures is less than 1: \(\frac{p_\mathrm{O}}{p_\mathrm{N}} = \frac{28}{32} < 1\) Which means: \(p_\mathrm{O} < p_\mathrm{N}\) The correct answer is c) \(p_{\mathrm{O}}<p_{\mathrm{N}}\).

Key concepts

Moles of Gas

When working with gases, the concept of moles is crucial as it helps us understand the quantity of gas in terms of molecules rather than weight. A mole is a basic unit in chemistry that allows us to count particles like atoms, molecules, and ions.
The number of moles of a gas is determined using the formula:

• \( n = \frac{m}{M} \)

Here, \(n\) represents the number of moles, \(m\) is the mass, and \(M\) is the molar mass.
In this problem, since the masses of both gases are equal, understanding the difference in moles becomes significant. Knowing the moles helps us understand how many molecules are present in the given amount of gas, enabling us to connect it to properties like pressure and temperature.

Molar Mass

Molar mass is a key factor that influences the behavior of gases, particularly in determining the number of moles. It is defined as the mass of one mole of a substance, usually expressed in grams per mole.
For oxygen, the molar mass is 32 g/mol, whereas for nitrogen, it's 28 g/mol. This difference impacts the number of moles, even when their masses are the same.

• A lower molar mass means more moles for the same mass.
  
• This results in different pressures for gases under identical conditions.
  

Using the Ideal Gas Law, understanding the molar mass allows us to make essential calculations about pressure changes between gases. It shows how gases with different molar masses react differently under similar circumstances.

Pressure Comparison

Comparing pressures of gases is a practical application of the Ideal Gas Law, expressed as \( PV = nRT \). This law helps relate pressure (\(P\)), volume (\(V\)), number of moles (\(n\)), universal gas constant (\(R\)), and temperature (\(T\)).
In this context, the formula can demonstrate how pressure changes with varying moles. Here, since the volume and temperature are constant, differences in the number of moles directly influence pressure.

• With our gases, we found using this principle that \(p_{\mathrm{O}} < p_{\mathrm{N}}\) or the pressure of oxygen is less than that of nitrogen.
• This tells us that gases in identical containers with the same mass but different compositions can have different pressures.
  

The fundamental takeaway is understanding that when all other components are constant, the pressure will inversely relate to the molar mass due to the differing moles of gas present.