Question

One hundred milliliters of liquid nitrogen with a mass of \(80.7 \mathrm{~g}\) is sealed inside a 2 - \(\mathrm{L}\) container. After the liquid nitrogen heats up and turns into a gas, what is the pressure inside the container? a) 0.05 atm b) 0.08 atm c) \(0.09 \mathrm{~atm}\) d) 9.1 atm e) 18 atm

Short answer

Answer: 35.3 atm

Step-by-step solution

Find the moles of nitrogen gas

First, we need to find the number of moles of nitrogen gas. We are given the mass of liquid nitrogen (80.7 g) and we know that the molar mass of nitrogen gas, N2, is approximately 28 g/mol. To find the number of moles, divide the mass by the molar mass: \(n = \frac{80.7 \mathrm{~g}}{28 \mathrm{~g/mol}} = 2.882 \mathrm{~moles}\)

Assume room temperature

We are not given the temperature in the problem, so we will assume it is at room temperature. Room temperature is approximately 25 °C, which is equal to 298 K.

Determine the ideal gas constant value

We need the ideal gas constant R in atm*L/mol*K. The value of R in these units is 0.0821 atm*L/mol*K.

Use the Ideal Gas Law equation to solve for pressure

We have the number of moles (\(n = 2.882 \mathrm{~moles}\)), the volume of the container (\(V = 2 \mathrm{~L}\)), the temperature (\(T = 298 \mathrm{~K}\)), and the ideal gas constant (\(R = 0.0821 \mathrm{~atm*L/mol*K}\)). Now, we can plug these values into the Ideal Gas Law equation: \(P = \frac{nRT}{V} = \frac{(2.882 \mathrm{~moles})(0.0821 \mathrm{~atm*L/mol*K})(298 \mathrm{~K})}{2 \mathrm{~L}}\)

Calculate the pressure

Calculate the pressure using the provided values: \(P = \frac{(2.882)(0.0821)(298)}{2} = 35.3 \mathrm{~atm}\) Comparing the calculated pressure with the given options, we find that none of them match our result. However, it is possible that the problem contains incorrect options or information.

Key concepts

Gas Pressure Calculation

Understanding how to calculate gas pressure is essential when working with gases in chemistry. Gas pressure is the force that the gas exerts on the walls of its container, and can be determined using the Ideal Gas Law. This law relates the pressure (P), volume (V), number of moles (n), temperature (T), and a constant (R) called the ideal gas constant.

To calculate the pressure of a gas, the Ideal Gas Law equation is used: \[\begin{equation} P = \frac{nRT}{V} \end{equation}\]In this formula, 'n' represents the number of moles of the gas, 'R' is the ideal gas constant which depends on the units used for pressure and volume, 'T' is the temperature in Kelvin, and 'V' is the volume of the gas.

When you're given the mass of a gas and its volume, as in our exercise, the first step is to convert the mass into moles by using the molar mass of the gas. Once you have the moles, you can use the Ideal Gas Law to find the pressure.

Molar Mass of Nitrogen

The molar mass of a substance, often termed molecular weight, is the mass in grams of one mole of that substance. The molar mass of nitrogen (N2) is particularly important in calculations involving the Ideal Gas Law. Nitrogen gas consists of diatomic molecules, meaning two nitrogen atoms (N) are bonded together, and the standard atomic weight of nitrogen is approximately 14.007 g/mol. Thus, to find the molar mass of nitrogen gas (N2), we simply double the atomic weight:

\[\begin{equation} \text{Molar mass of N2} = 2 \times 14.007 \text{ g/mol} = 28.014 \text{ g/mol} \end{equation}\]
As seen in our exercise, the molar mass allows us to convert the mass of nitrogen in grams to the amount in moles, which is a step necessary for using the Ideal Gas Law. The correct calculation and understanding of the molar mass are critical for solving problems involving gas properties.

Ideal Gas Constant

The ideal gas constant, symbolized as 'R', is a proportionality constant that appears in the Ideal Gas Law equation. It connects the physical properties of an ideal gas, namely pressure, volume, temperature, and the number of moles. The value of R changes depending on the units used for these gas properties. In our exercise, we use the value of R that corresponds with atmospheres for pressure, liters for volume, moles for the quantity of gas, and Kelvin for temperature:

\[\begin{equation} R = 0.0821 \frac{\text{atm} \cdot L}{\text{mol} \cdot K} \end{equation}\]
The ideal gas constant is derived from the equation of state for an ideal gas, which assumes that gas particles do not attract or repel each other and take up negligible space. While real gases do not perfectly follow these assumptions, the ideal gas law provides a useful approximation for gas behavior under many conditions. The value of R is key to connecting all variables in the Ideal Gas law, allowing for the calculation of unknowns such as gas pressure in the container.