Question

Imagine designing an experiment in which the presence of a gas is determined by simply listening to the gas with your ear. The human ear can detect pressures as low as \(2 \times 10^{-5} \mathrm{N} \mathrm{m}^{-2}\). Assuming that the eardrum has an area of roughly \(1 \mathrm{mm}^{2},\) what is the minimum collisional rate that can be detected by ear? Assume that the gas of interest is \(\mathrm{N}_{2}\) at \(298 \mathrm{K}\).

Short answer

The minimum collisional rate that can be detected by ear is approximately \(1.04 \times 10^{-11}\) collisions per second. This is calculated by first finding the force detectable by the human ear, then using the ideal gas law to find the number of moles in the gas, and finally calculating the collisional rate using the formula: Collisional Rate = Pressure / (n × R × T).

Step-by-step solution

Calculate the Force

First, we need to find the force that the human ear can detect. Using the formula Pressure = Force/Area, we can rearrange to find the force: Force = Pressure × Area Given the pressure threshold is \(2 \times 10^{-5} \mathrm{N} \mathrm{m}^{-2}\), and the eardrum has an area of roughly \(1 \mathrm{mm}^{2}\), which is \(1 \times 10^{-6} \mathrm{m}^{2}\) in SI units. Force = \( (2 \times 10^{-5} \mathrm{N} \mathrm{m}^{-2}) (1 \times 10^{-6} \mathrm{m}^{2}) \) Force = \( 2 \times 10^{-11} \mathrm{N} \)

Calculate the Number of Moles Using Ideal Gas Law

We will now use the ideal gas law to find the number of moles (n) in the gas. This will help us determine the collisional rate. The ideal gas law is: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature in Kelvin. It's given that the nitrogen gas (\(\mathrm{N}_2\)) is at a temperature of \(298 \mathrm{K}\). But we need to find the number of moles based on the minimum force detectable by the human ear. To do this, we should divide the force by the Avogadro's number (6.022 × 10²³ molecules/mol). Number of moles, n = Force / (Avogadro's number × mass of one nitrogen molecule) n = \( \dfrac{2 \times 10^{-11} \mathrm{N}}{(6.022 \times 10^{23} \text{ molecules/mol})(4.65 \times 10^{-26} \text{ kg)} \) n = \( \dfrac{2 \times 10^{-11} \mathrm{N}}{(6.022 \times 10^{23} \text{ molecules/mol})(4.65 \times 10^{-26} \text{ kg)} \) n ≈ 0.071 mol

Calculate the Collisional Rate

Now that we have the number of moles (n), we can calculate the collisional rate using the formula: Collisional Rate = Pressure / (n × R × T) Where P is the pressure, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. Collisional Rate = \( \dfrac{2 \times 10^{-5} \mathrm{N} \mathrm{m}^{-2}}{(0.071 \text{ mol})(8.314 \text{ J/mol·K})(298 \text{ K})} \) Collisional Rate ≈ 1.04 × 10⁻¹¹ s⁻¹ So, the minimum collisional rate that can be detected by ear is approximately 1.04 × 10⁻¹¹ collisions per second.

Key concepts

Understanding Gas Pressure Detection

Gas pressure detection plays a critical role in many scientific applications, including the experimental scenario where one listens to gas with the ear to detect its presence. At the foundation of this intriguing idea is the understanding that pressure is essentially the continuous physical force exerted on or against an object by something in contact with it—in this case, gas molecules.

This force per unit area can be perceived by the human eardrum as sound when the molecular collisions of the gas are frequent and strong enough. Our eardrums are sensitive membranes capable of detecting even minute pressures which can be translated into sound frequencies. This sensitivity threshold means that, under the right circumstances, one could indeed 'hear' the presence of a gas based on its collisional interactions with the eardrum.

The Ideal Gas Law in Action

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and number of moles of an ideal gas. Traditionally represented as PV = nRT, the ideal gas law allows us to explore the behavior of gases and make pertinent calculations relevant to our daily lives and scientific endeavors.

In the context of our exercise, we apply this law to define the relationship between the detectable force on the eardrum, the pressure exerted by the gas, and subsequently, determine the number of moles of gas molecules responsible for this force. This application underscores the versatility of the ideal gas law and its central role in explaining the behavior of gases under various conditions.

Eardrum Sensitivity Threshold

The human ear is a remarkable organ, capable of detecting a remarkable range of sounds from the faintest whisper to the loudest roar. This ability is deeply connected to the sensitivity threshold of the eardrum. In the scientific scenario of using our hearing to detect gas presence, it translates to the human ear being able to pick up on the slightest changes in pressure, as delicate as 2 x 10^-5 N/m^2.

It's this incredible sensitivity that could, theoretically, allow us to detect the collision rate of nitrogen molecules within a gas at room temperature. When we talk about this threshold in terms of physics, we're essentially discussing the minimal amount of force required to create a sound wave that the eardrum can convert into an audible signal for our brains to process.

Utilizing Avogadro's Number

Avogadro's number, approximately 6.022 x 10^23, represents the number of units in one mole of a substance and is a key component in chemistry known for its use in converting between microscopic and macroscopic scales. Its utilization in the exercise is vital for calculating the number of nitrogen molecules exerting force on the eardrum, and subsequently, the moles of gas involved.

By including Avogadro's number in our calculations, we bridge the gap between observable physical phenomena, such as audible sound, and the molecular components responsible for them. It allows us not only to count the number of molecules but also to understand the direct impact of molecular collisions on our sensory experiences, such as hearing.