Question

A sealed container contains 1.00 mole of neon gas at STP. Estimate the number of neon atoms having speeds in the range from \(200.00 \mathrm{~m} / \mathrm{s}\) to \(202.00 \mathrm{~m} / \mathrm{s}\). (Hint: Assume that the probability of neon atoms having speeds between \(200.00 \mathrm{~m} / \mathrm{s}\) and \(202.00 \mathrm{~m} / \mathrm{s}\) is constant.

Short answer

Question: Estimate the number of neon atoms in a gas sample that have speeds between 200.00 m/s and 202.00 m/s, assuming STP for neon gas. Answer: To estimate the number of neon atoms, we need to calculate the average probability density (\(\bar{p}\)) within the given speed range using the Maxwell-Boltzmann distribution. With the given temperature (273.15 K) and neon atom mass (3.35 * 10^{-26} kg/atom), we can calculate the probability densities at 200.00 m/s and 202.00 m/s and take their average. Finally, multiply the average probability density by the total number of particles and the change in speed to estimate the number of neon atoms within the given speed range.

Step-by-step solution

Understand the Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds for particles in an ideal gas. The PDF of the distribution is given by: \(p(v) = 4\pi \cdot (\frac{m}{2\pi k_BT})^{3/2} \cdot v^2 \cdot e^{-\frac{mv^2}{2k_BT}}\) where - \(p(v)\) is the probability density of a particle having velocity \(v\) - \(m\) is the mass of a particle - \(k_B\) is Boltzmann's constant (\(1.38 * 10^{-23} \mathrm{~J} / \mathrm{K}\)) - \(T\) is the temperature in Kelvin - \(v\) is particle's velocity in m/s

Calculate mass of a neon atom

To determine the mass of a neon atom, use its molar mass and convert it to the mass of an individual atom using Avogadro's number (\(6.022 \times 10^{23} \mathrm{atoms/mol}\)). The molar mass of neon gas is 20.18 g/mol, so the mass of a neon atom is: \(m =\) (20.18 g/mol) / (6.022 * 10^{23} atoms/mol) = 3.35 * 10^{-26} \mathrm{~kg/atom}$.

Calculate the temperature in Kelvin

As we are given STP, the temperature is 273.15 K.

Probability density function for given speeds

Calculate the PDF for both velocities, 200.00 m/s and 202.00 m/s, for neon atoms utilizing the Maxwell-Boltzmann distribution. To simplify, first calculate the constant factor \(A = 4\pi \cdot (\frac{m}{2\pi k_BT})^{3/2}\) and then compute the PDF for both velocities as: \(p(v) = A \cdot v^2 \cdot e^{-\frac{mv^2}{2k_BT}}\) For v = 200.00 m/s, and v = 202.00 m/s, the calculated probability densities will be referred to as \(p(200)\) and \(p(202)\), respectively.

Calculate average probability density

We will assume the probability density to be constant between the given velocities. Hence, the average probability density within the given range will be the average of \(p(200)\) and \(p(202)\). \(\bar{p} = \frac{p(200) + p(202)}{2}\)

Estimate the number of atoms in the given speed range

Since we have the average probability density in the range, we need to calculate the number of atoms in the given speed range. To do this, the average probability density is multiplied by the total number of particles and the change in speed: \(n_{atoms} = (1.00 \mathrm{mol} \cdot 6.022\times10^{23}\mathrm{atoms/mol}) \times \bar{p} \times (202.00 - 200.00 \mathrm{~m} / \mathrm{s})\) Now, calculate the value of \(n_{atoms}\) to obtain the required estimation.

Key concepts

Neon Gas Particle Speeds

When studying gases like neon under certain conditions, it's fascinating to consider how the individual particles move. Although we can't track each one, we can predict their behavior using statistical mechanics. The speeds of these particles are not uniform but adhere to the Maxwell-Boltzmann distribution. This distribution reveals that at any given temperature, there's a wide spread of velocities—some particles may be moving very slowly, while others could be zooming at high speeds. The remarkable aspect here is that we can define the most probable speed, average speed, and root-mean-square speed for these particles.

In our context, we're specifically interested in neon atoms within a tight speed range around 200 m/s, which is rather specific compared to the entire distribution. Even though this speed range is quite narrow, we still expect to find a significant number of atoms within it due to the sheer number of particles involved.

Boltzmann's Constant

One of the cornerstone concepts in understanding the behavior of gases is Boltzmann's constant, denoted by \(k_B\). This fundamental physical constant connects macroscopic and microscopic physics by relating temperature (a property we can feel and measure on the large scale) with kinetic energy on the atomic scale.

It has a value of approximately \(1.38 \times 10^{-23} J/K\) and plays a pivotal role in the Maxwell-Boltzmann distribution, which is used to describe gas particle behavior. When paired with the particle's mass and the temperature, Boltzmann's constant helps us calculate the probability of finding a particle at a certain speed.

Probability Density Function

A probability density function (PDF) is a fundamental concept in statistics and physics, representing how the probabilities of a continuous random variable are distributed across different values. For neon gas particles, the Maxwell-Boltzmann distribution acts as the PDF for the speed of particles.

This function's graph peaks at the most probable speed and then tapers off, signifying that there are fewer particles at very high or very low speeds. By calculating the PDF for specific speeds, we can estimate how many particles are likely to be found in that narrow speed range.

STP Conditions

STP stands for Standard Temperature and Pressure, which are typically used as standard conditions for experiments and calculations in chemistry and physics. Standard temperature is defined as 0°C or 273.15 K, and standard pressure is set at 1 atmosphere. These conditions provide a common reference, making it easier to compare and reproduce experimental results.

When gases are described under STP, it means their behavior is predicted under these agreed-upon conditions, and the Maxwell-Boltzmann distribution would be calculated using the STP temperature of 273.15 K.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry with the impressive value of \(6.022 \times 10^{23}\) entities per mole. This staggering number indicates the number of atoms or molecules in one mole of any substance, tying in macroscopic qualities (like mass of a mole measured in grams) with the microscopic world (individual atoms or molecules).

By using Avogadro's number, we connect the weight of neon's molar mass to the mass of a single neon atom. Avogadro's number is crucial for calculations in our exercise as it helps determine the total number of particles we are dealing with when we talk about 1 mole of neon gas.