Question

At what temperature does \(100 .\) mL of \(\mathrm{N}_{2}\) at \(300 . \mathrm{K}\) and 1.13 atm occupy a volume of \(500 .\) mL at a pressure of 1.89 atm?

Short answer

The temperature at which 100 mL of N2 gas at 300 K and 1.13 atm occupies a volume of 500 mL at a pressure of 1.89 atm is approximately 797 K.

Step-by-step solution

Understand the Ideal Gas Law

The Ideal Gas Law is given by the equation: \(PV = nRT\) where: P = pressure V = volume n = moles of gas R = universal gas constant (\(8.314 \frac{J}{mol \cdot K}\)) T = temperature in Kelvin

Calculate the number of moles of gas

To find the number of moles of gas, we can rearrange the Ideal Gas Law formula: \(n = \frac{PV}{RT}\) Using the initial conditions: P = 1.13 atm V = 100 mL T = 300 K R = 0.08206 \(\frac{atm \cdot L}{mol \cdot K}\) (since our values are in atm and mL) \(n = \frac{(1.13)(0.100)}{(0.08206)(300)} = 0.00459 \, mol\) We now know the number of moles of nitrogen gas is 0.00459 mol.

Understand the Combined Gas Law

The Combined Gas Law is given by the equation: \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) where: \(P_1\) and \(P_2\) are the initial and final pressures \(V_1\) and \(V_2\) are the initial and final volumes \(T_1\) and \(T_2\) are the initial and final temperatures

Solve for the final temperature \(T_2\)

To find the final temperature, we can use the Combined Gas Law equation and the values provided: \(P_1 = 1.13 \, atm\) \(V_1 = 100 \, mL\) \(T_1 = 300 \, K\) \(P_2 = 1.89 \, atm\) \(V_2 = 500 \, mL\) Rearrange the Combined Gas Law equation to solve for \(T_2\): \(T_2 = \frac{P_2V_2T_1}{P_1V_1} = \frac{(1.89)(0.500)(300)}{(1.13)(0.100)} = 796.99 \, K\)

Round the final answer

In this exercise, we should round our answer to reasonable decimal places for practical purposes. So, \(T_2\) is approximately 797 K. As requested, the temperature at which 100 mL of N2 gas at 300 K and 1.13 atm occupies a volume of 500 mL at a pressure of 1.89 atm is approximately 797 K.

Key concepts

Combined Gas Law

The Combined Gas Law is a powerful tool for understanding how gases behave under various conditions. It combines three individual gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. This law allows you to compare two different sets of conditions for a gas. By using the equation \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\), you can predict how a gas will react if the pressure, volume, or temperature changes.
The variables are:

• \(P_1\) and \(P_2\): initial and final pressures
• \(V_1\) and \(V_2\): initial and final volumes
• \(T_1\) and \(T_2\): initial and final temperatures in Kelvin

One important point to remember about the Combined Gas Law is that it assumes the number of moles of gas remains constant throughout the process.
This law is especially useful when one of these variables changes while the others remain constant, such as when heating or compressing a gas.

moles of gas

In the context of gases, the concept of moles is crucial because it ties into how we quantify and predict gas behavior. A mole is a unit that defines a quantity of substances in terms of the number of molecules or atoms. In gas equations like the Ideal Gas Law, the term \(n\) represents moles of gas.
Moles can be calculated using the rearranged Ideal Gas Law: \(n = \frac{PV}{RT}\). Here, you need to consider:

• Pressure \(P\)
• Volume \(V\)
• Temperature in Kelvin \(T\)
• Universal gas constant \(R\) (adjust to units such as atm and L when necessary)

In the exercise, by applying these principles, you find that there are approximately 0.00459 moles of nitrogen gas present. Knowing the moles helps in predicting what will happen when conditions change.

pressure-volume relationship

The pressure-volume relationship in gases is described in part by Boyle's Law, which is integrated into the Combined Gas Law. According to Boyle's Law, pressure and volume have an inverse relationship when temperature and moles of gas are constant. This means that:

• When pressure increases, volume decreases
• When volume increases, pressure decreases

This relationship can help us understand gas dynamics in enclosed systems, like a balloon compressing under higher outside pressure. In the context of our exercise, initial conditions of pressure (1.13 atm) and volume (100 mL) are altered, impacting final conditions of the gas, like occupancy of new volume (500 mL) at different pressure (1.89 atm). Understanding these relationships helps calculate the new state of the gas accurately.

temperature in Kelvin

Temperature is a fundamental aspect when discussing gases, particularly in relation to their kinetic energy and movement. In gas calculations, temperature must be measured in Kelvin. The Kelvin scale is important because it starts at absolute zero, meaning temperatures are always positive, which simplifies calculations.
The formula for converting Celsius to Kelvin is: \[K = °C + 273.15\]
Kelvin is used so that the directly proportional relationships in gas laws, like Charles's Law and the Combined Gas Law, are mathematically consistent.
In the original exercise, the initial temperature was 300 K. When calculating the final temperature with the Combined Gas Law, it's determined to be approximately 797 K. This significant change reflects increased thermal energy required to occupy more space at higher pressure. Thus, Kelvin ensures that our mathematical treatment of gas behavior is consistent and accurate.