Problem 22 - An Introduction to Thermal Physics

Question

If you poke a hole in a container full of gas, the gas will start leaking out. In this problem you will make a rough estimate of the rate at which gas escapes through a hole. (This process is called effusion, at least when the hole is sufficiently small.) (a) Consider a small portion (area \(=A\) ) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval \(\Delta t\) is \(P A \Delta t /(2 m \overline{v_{x}}),\) where \(P\) is the pressure, \(m\) is the average molecular mass, and \(\overline{v_{x}}\) is the average \(x\) velocity of those molecules that collide with the wall. (b) It's not easy to calculate \(\overline{v_{x}},\) but a good enough approximation is \((\overline{v_{x}^{2}})^{1 / 2}\) where the bar now represents an average over all molecules in the gas. Show that \((\overline{v_{x}^{2}})^{1 / 2}=\sqrt{k T / m}\) (c) If we now take away this small part of the wall of the container, the molecules that would have collided with it will instead escape through the hole. Assuming that nothing enters through the hole, show that the number \(N\) of molecules inside the container as a function of time is governed by the differential equation $$\frac{d N}{d t}=-\frac{A}{2 V} \sqrt{\frac{k T}{m}} N$$ Solve this equation (assuming constant temperature) to obtain a formula of the form \(N(t)=N(0) e^{-t / \tau},\) where \(\tau\) is the "characteristic time" for \(N\) \((\text { and } P)\) to drop by a factor of \(e\) (d) Calculate the characteristic time for a gas to escape from a 1 -liter container punctured by a \(1-\mathrm{mm}^{2}\) hole. (e) Your bicycle tire has a slow leak, so that it goes flat within about an hour after being inflated. Roughly how big is the hole? (Use any reasonable estimate for the volume of the tire.) (f) In Jules Verne's Round the Moon, the space travelers dispose of a dog's corpse by quickly opening a window, tossing it out, and closing the window. Do you think they can do this quickly enough to prevent a significant amount of air from escaping? Justify your answer with some rough estimates and calculations.

Step-by-Step Solution

Answer

The rate of effusion depends on the container volume and hole size. Characteristic escape time can be calculated using these parameters. The size of a hole required for a tire to deflate in an hour can also be estimated.

Step 1: Calculate Molecules Colliding with Surface

We start by considering a small surface area \(A\) on the inside of a container. The number of molecules hitting this area in a time \(\Delta t\) can be found by considering the pressure \(P\), the change in time \(\Delta t\), the molecule mass \(m\), and the average x-velocity \(\overline{v_x}\). The formula for pressure \(P\) can be given as \(P = \frac{F}{A}\), where \(F\) is the force. The force exerted by molecules is also related to the average x-velocity and number of molecules colliding. Thus, we derive:\[ \text{Number of Molecules} = \frac{P A \Delta t}{2 m \overline{v_x}} \]

Step 2: Approximate Average X-Velocity

Here, we approximate the average velocity of molecules moving in the x-direction, \(\overline{v_x}\), as the root mean square velocity, \((\overline{v_x^2})^{1/2}\), because it's simpler to compute. To find this approximation, we use kinetic theory, where the root mean square speed \(v_{rms}\) is given by:\[ v_{rms} = \sqrt{\overline{v_x^2}} = \sqrt{\frac{kT}{m}} \]where \(k\) is Boltzmann's constant and \(T\) is temperature.

Step 3: Derive the Differential Equation for Effusion

Using our previous findings, we consider a small area of the wall being removed. The number of molecules escaping is proportional to the number of molecules colliding, previously given by:\[ \frac{dN}{dt} = -\frac{A}{2V}\sqrt{\frac{kT}{m}}N \]Here, \(-\) sign indicates a decrease in the number as the molecules are leaving the container. This differential equation describes the rate at which the number of molecules \(N\) decreases.

Step 4: Solve the Differential Equation

The separation of variables technique solves the equation \(\frac{dN}{dt} = -\frac{A}{2V}\sqrt{\frac{kT}{m}}N\). Dividing both sides by \(N\) and integrating both sides over \(t\), we obtain:\(\ln N = -\frac{A}{2V}\sqrt{\frac{kT}{m}} t + C\).By exponentiating, we find the solution:\[ N(t) = N(0)e^{-t/\tau} \]where \( \tau = \frac{2V}{A}\sqrt{\frac{m}{kT}} \) is the characteristic time.

Step 5: Calculate the Characteristic Time for Gas Escape

To find the characteristic time \(\tau\) for a 1-liter container with a 1 mm² hole, convert all measures into SI units: volume \(V = 1\times10^{-3} \text{ m}^3\) and area \(A = 1\times10^{-6} \text{ m}^2\). Using air's average molecular mass \(m\) and temperature \(T\), calculate \(\tau\) using:\[ \tau = \frac{2V}{A}\sqrt{\frac{m}{kT}} \]

Step 6: Estimate the Size of the Tire Hole

Assuming the volume of a bicycle tire, say 3 liters, calculate the hole size given \(\tau = 1 \text{ hr}\). Rearranging the formula for \(\tau\), find the appropriate \(A\) in:\[ \tau = \frac{2V}{A}\sqrt{\frac{m}{kT}} \].

Step 7: Assess Window Opening Scenario

Estimate the time to open and close a window, e.g., 1 second. Using \(V\approx100 L\) and a reasonable opening size, calculate \(\tau\). Compare \(\tau\) with the time to assess significant air loss:\[ \text{Air Loss} = 1 - e^{-t/\tau} \].

Key Concepts

Kinetic Theory of Gases

The Kinetic Theory of Gases is a fundamental framework in physics that helps explain the behavior of gases. It assumes that a gas comprises many small particles, usually molecules or atoms, which are constantly and randomly moving in all directions. This theory is critical because it provides a link between the macroscopic properties of gases, like pressure and temperature, and the microscopic behavior of molecules.
Understanding the random motion of molecules helps predict and explain macroscopic phenomena.

• Gas Pressure: Pressure results from the collective force exerted by gas molecules colliding with the walls of their container. According to the kinetic theory, pressure is directly related to the kinetic energy of gas molecules.
• Temperature: From this viewpoint, temperature is a measure of the average kinetic energy of the gas molecules. Higher temperatures mean faster-moving particles.
• Volume and Moles: The theory also examines how the volume of the gas container and the number of gas moles affect the behavior of gases.

Combining these insights helps explain why gases expand when heated and how gas pressure increases when the volume of the container decreases.

Root Mean Square Speed

The Root Mean Square Speed is a measure used to determine the speed of particles in a gas. Instead of looking at individual particle speeds, it provides an average speed for all particles in the gas.
It is vital because the individual speeds vary widely, and using an average provides simplicity and practicality. The root mean square speed is defined as the square root of the average of the squared speeds of gas molecules.
In mathematical terms:\[ v_{rms} = \sqrt{\overline{v_x^2}} = \sqrt{\frac{kT}{m}}\]Where:

• \(k\) is Boltzmann's constant: It aids in relating temperature with energy at a molecular level.
• \(T\) is the absolute temperature of the gas.
• \(m\) is the mass of a single molecule.

By inserting the temperature and molecular mass, one can determine the average speed of molecules in a gas sample.

Differential Equation for Gas

In the context of effusion, the differential equation describes how the number of gas molecules in a container changes over time. When a small section of a container wall is removed, molecules that would have collided with it escape instead.
This process is mathematically represented by:\[\frac{dN}{dt} = -\frac{A}{2V}\sqrt{\frac{kT}{m}}N\]The equation captures the rate of decrease in the number of molecules. Here's what each element signifies:

• \(dN/dt\) represents the rate of change of the number of molecules with respect to time.
• \(A\) is the area of the hole created by removing part of the wall.
• \(V\) is the volume of the container.
• The root mean square term, \(\sqrt{\frac{kT}{m}}\), reflects the average speed of molecules trying to escape.

The negative sign signifies that the number of molecules is decreasing as gas effuses through the hole. Solving this equation gives insights into how long it takes for the gas to exit completely.

Escape Rate of Gases

The escape rate of gases, also known as effusion, involves understanding how quickly gas molecules leave a container through a small opening. Effusion is influenced by several factors, including the gas temperature, molecular weight of molecules, and the size of the hole.
Effusion is differentiated from diffusion, which deals with gas movement across containers with no net boundary. Some key points to consider include:

• Lighter gas molecules effuse faster due to their higher speed at any given temperature compared to heavier molecules.
• Higher temperatures increase molecular speeds, thus increasing effusion rates.
• Graham's Law of Effusion: It states that the rate of effusion is inversely proportional to the square root of the molar mass of the gas.

By understanding these factors, one can predict how fast or slow a gas will leak out of a punctured container or sealed environment.