Nitrogen Gas Calculations: Boyle's Law And Molar Mass

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Hey guys! Let's dive into some nitrogen gas calculations. We've got a cylindrical bottle filled with nitrogen, and we're going to explore some cool physics concepts like Boyle-Mariotte's Law and molar mass. So, let's get started!

1. Stating Boyle-Mariotte's Law

Okay, so what's Boyle-Mariotte's Law all about? In simple terms, Boyle-Mariotte's Law describes the relationship between the pressure and volume of a gas when the temperature and the amount of gas are kept constant. Think of it like squeezing a balloon – as you decrease the volume, the pressure inside increases, and vice versa. This inverse relationship is at the heart of the law. You see, this principle is crucial in understanding how gases behave under different conditions, making it applicable in fields ranging from chemistry to engineering.

Mathematically, we can express Boyle-Mariotte's Law as:

  • P₁V₁ = P₂V₂

Where:

  • P₁ is the initial pressure.
  • V₁ is the initial volume.
  • P₂ is the final pressure.
  • V₂ is the final volume.

This equation tells us that the product of pressure and volume remains constant as long as the temperature and the amount of gas don't change. It's a pretty handy tool for predicting how a gas will behave in different situations. For instance, if you know the initial pressure and volume of a gas, and you change the volume, you can easily calculate the new pressure using this formula. It's like having a superpower for gas calculations!

The law is named after Robert Boyle and Edme Mariotte, who independently discovered this relationship in the 17th century. Boyle, an Irish chemist and physicist, published his findings in 1662, while Mariotte, a French physicist and priest, published his work in 1676. It's a classic example of scientific discovery where two people, working separately, arrived at the same conclusion. This just goes to show that scientific truths often emerge when the time is right, and multiple minds are exploring the same questions. This principle has practical applications in various fields, such as scuba diving, where understanding the relationship between pressure and volume is crucial for safe diving practices. In industrial processes, Boyle-Mariotte's Law helps in designing and operating equipment that handles gases, ensuring efficiency and safety.

In our specific scenario with the nitrogen gas in the cylindrical bottle, Boyle-Mariotte's Law helps us understand how the pressure inside the bottle would change if we were to change the volume. If we compressed the gas, the pressure would increase, and if we expanded the volume, the pressure would decrease. This fundamental principle is the foundation for many gas-related calculations and applications.

2. Calculating the Molar Mass of Nitrogen Gas

Next up, let's figure out the molar mass of nitrogen gas (N₂). What's molar mass, you ask? Well, it's the mass of one mole of a substance. A mole is just a specific number of molecules (6.022 x 10²³ molecules, also known as Avogadro's number), so the molar mass tells us how much one "mole-sized" group of those molecules would weigh. To calculate the molar mass of N₂, we need to know the atomic mass of nitrogen (N).

The atomic mass of nitrogen is approximately 14.01 atomic mass units (amu). You can usually find this value on the periodic table. Since nitrogen gas exists as a diatomic molecule (N₂), meaning two nitrogen atoms are bonded together, we need to multiply the atomic mass of nitrogen by 2 to get the molar mass of N₂.

So, the calculation is:

  • Molar mass of N₂ = 2 * 14.01 amu = 28.02 g/mol

Therefore, the molar mass of nitrogen gas is approximately 28.02 grams per mole. This value is super important because it allows us to convert between the mass of nitrogen gas and the number of moles, which is essential for many chemical calculations. For instance, if you know you have 56.04 grams of nitrogen gas, you can divide that by the molar mass (28.02 g/mol) to find out you have 2 moles of N₂.

Understanding the molar mass is crucial in various applications, such as determining the amount of reactants needed in a chemical reaction or calculating the density of a gas. In the context of our cylindrical bottle, the molar mass helps us relate the mass of the nitrogen gas to the number of molecules present, which is a key factor in understanding the gas's behavior under different conditions. The molar mass calculation can also be considered a fundamental step in quantitative chemistry, where precise measurements and calculations are essential for accurate results. In environmental science, for instance, knowing the molar mass of gases like nitrogen oxides is vital for assessing air quality and pollution levels.

3. Calculating the Quantity of...

To continue with our exploration of the nitrogen gas in the cylindrical bottle, we need to figure out what specific quantity we're trying to calculate. The original prompt ends somewhat abruptly, so let's consider a few possibilities and then delve into how we'd calculate them.

Option 1: Calculating the Number of Moles

Perhaps we want to find out how many moles of nitrogen gas are present in the bottle. To do this, we'll need to use the ideal gas law and the information we have: the volume (V = 0.7 m³), the pressure (P = 4.3 hPa), and the temperature (T = 15°C).

The ideal gas law is expressed as:

  • PV = nRT

Where:

  • P is the pressure.
  • V is the volume.
  • n is the number of moles (what we want to find).
  • R is the ideal gas constant (8.314 J/(mol·K)).
  • T is the temperature in Kelvin.

First, we need to convert the pressure from hectopascals (hPa) to Pascals (Pa) and the temperature from Celsius (°C) to Kelvin (K):

  • Pressure: 4.3 hPa = 4.3 * 100 Pa = 430 Pa
  • Temperature: 15°C = 15 + 273.15 K = 288.15 K

Now, we can rearrange the ideal gas law to solve for n:

  • n = PV / RT

Plugging in the values:

  • n = (430 Pa * 0.7 m³) / (8.314 J/(mol·K) * 288.15 K)
  • n ≈ 0.126 moles

So, there are approximately 0.126 moles of nitrogen gas in the bottle. This calculation is crucial in various applications, including chemical reactions and industrial processes, where knowing the exact amount of reactants is essential for achieving desired outcomes. In the field of materials science, for instance, the precise control of gas quantities is critical in processes like chemical vapor deposition, where thin films are grown by reacting gaseous precursors.

Option 2: Calculating the Mass of Nitrogen Gas

Another possibility is that we want to calculate the mass of nitrogen gas in the bottle. Now that we know the number of moles (n ≈ 0.126 moles) and the molar mass of N₂ (28.02 g/mol), we can easily calculate the mass using the following formula:

  • Mass = n * Molar mass
  • Mass = 0.126 moles * 28.02 g/mol
  • Mass ≈ 3.53 grams

Therefore, there are approximately 3.53 grams of nitrogen gas in the bottle. This calculation highlights the importance of understanding the relationship between moles, molar mass, and mass, which is a fundamental concept in chemistry. Knowing the mass of gas is essential in many practical applications, such as determining the storage capacity of gas cylinders or calculating the amount of gas released in a chemical reaction.

Option 3: Calculating the Density of Nitrogen Gas

Yet another possibility is that we want to calculate the density of the nitrogen gas in the bottle. Density is defined as mass per unit volume, and we already know both the mass (3.53 grams) and the volume (0.7 m³). However, to get a more standard unit for density, let's convert the volume from cubic meters to liters:

  • 0.7 m³ = 0.7 * 1000 liters = 700 liters

Now, we also need to convert the mass from grams to kilograms:

  • 3.53 grams = 3.53 / 1000 kg = 0.00353 kg

Now we can calculate the density:

  • Density = Mass / Volume
  • Density = 0.00353 kg / 0.7 m³
  • Density ≈ 0.00504 kg/m³

Or, if we want to express the density in grams per liter:

  • Density = 3.53 grams / 700 liters
  • Density ≈ 0.00504 g/L

So, the density of the nitrogen gas in the bottle is approximately 0.00504 kg/m³ or 0.00504 g/L. This value provides insight into how much gas is packed into the given volume, which is particularly relevant in industrial settings where gases are stored and transported.

Conclusion

So, guys, we've covered a lot about nitrogen gas! We revisited Boyle-Mariotte's Law, calculated the molar mass of nitrogen gas, and explored different ways to calculate the quantity of nitrogen gas in a cylindrical bottle. Whether it's finding the number of moles, the mass, or the density, these calculations are super useful in various scientific and industrial applications. Keep exploring and stay curious!