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Chapter 12: Problem 16

A vessel of volume \(V\) contains a gas that is kept at constant temperature. The gas slowly leaks out of a small hole of area \(A\). The outside pressure is so low that no atoms leak back. (a) Prove that the pressure at any time \(t\) is given by $$ P=P_{0} e^{-k^{\prime} t} $$ where \(P_{0}\) is the initial pressure.

Short Answer

Expert verified
Pressure is given by \( P = P_{0} e^{-k't} \).

Step by step solution

01

- Understand the problem

We need to show the pressure of a gas in a vessel as a function of time, given that the gas leaks out of a small hole and the outside pressure is negligible.
02

- Relate pressure and volume

Start by recalling that for an ideal gas, pressure is related to the number of moles of gas (n) and volume (V) by the ideal gas law: \[ P = \frac{nRT}{V} \]
03

- Determine the rate of gas escaping

Denote the number of moles of gas remaining in the vessel at time \( t \) as \( n(t) \). The rate at which gas escapes through the hole can be expressed as: \[ \frac{dn}{dt} = -k n(t) \] where \( k \) is a constant that depends on the size of the hole (area A) and other physical properties.
04

- Solve the differential equation

The equation \( \frac{dn}{dt} = -k n(t) \) is a first order linear differential equation. Solving this gives: \[ n(t) = n(0) e^{-kt} \] where \( n(0) \) is the initial number of moles.
05

- Relate initial conditions

Since initial pressure \( P_{0} \) is given at \( t = 0 \), use the ideal gas law: \[ P_{0} = \frac{n(0)RT}{V} \]
06

- Express pressure as a function of time

From the ideal gas law, pressure at any time \( t \) is: \[ P(t) = \frac{n(t)RT}{V} = \frac{n(0)RT}{V} e^{-kt} = P_{0} e^{-kt} \]
07

- Introduce problem constants

Define a new constant \( k' = k \) for simplification, resulting in the final expression: \[ P(t) = P_{0} e^{-k't} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gas Pressure and Volume Relationship
The relationship between gas pressure and volume is explained by the Ideal Gas Law. This law states that for an ideal gas, pressure (P) is related to the number of moles (n), the gas constant (R), and temperature (T), in relation to the container's volume (V). The equation is: \[ P = \frac{nRT}{V} \]Key points to remember:
  • Pressure is directly proportional to the number of gas molecules and the absolute temperature.
  • Pressure is inversely proportional to the volume of the container.
If you change one of these variables, the others must adjust to maintain the equation equilibrium. This fundamental relationship helps in understanding how pressure changes as gas escapes from the vessel.
Rate of Gas Leakage
In the exercise, we deal with the situation where gas leaks out of a vessel. The rate of gas leakage can be described using a differential equation, which provides insight into how quickly the gas escapes. The specific relationship is: \[ \frac{dn}{dt} = -kn(t) \]Here,
  • \(\frac{dn}{dt}\): The rate of change of the number of moles of gas over time.
  • \(k\): A constant that depends on the hole's area (A) and other physical properties.
  • \(n(t)\): The number of moles of gas remaining at time \(t\).
The negative sign indicates that the number of moles decreases over time. This differential equation captures the rate at which gas leaks from the vessel.
First Order Linear Differential Equation
A first order linear differential equation is used to describe the rate of changes in various scenarios, such as gas leakage. In our exercise, the differential equation \[ \frac{dn}{dt} = -kn(t) \] falls under this category.This type of equation is characterized by:
  • Being linear, meaning the dependent variable (n) and its derivative \(\frac{dn}{dt}\) appear to the first power.
  • Involving only the first derivative.
To solve the equation, we separate the variables and integrate both sides:\[ \int \frac{1}{n} dn = -k \int dt \]This yields:\[ \text{ln}(n) = -kt + C \]By exponentiating both sides, we find:\[ n(t) = n(0) e^{-kt} \]This demonstrates the exponential decay of the gas quantity over time.
Exponential Decay in Gas Pressure
Exponential decay describes processes that reduce quantities at rates proportional to their current value. In our exercise, the pressure of the gas in the vessel decays exponentially because gas leaks out over time. This is expressed with:\[ P(t) = P_{0} e^{-k't} \]Where:
  • \(P(t)\): Pressure at time \(t\).
  • \(P_{0}\): Initial pressure.
  • \(k'\): A constant based on the leakage rate.
Exponentially decaying functions have the form \(e^{-kt}\), indicating rapid initial change that slows over time. This principle helps predict how quickly pressure decreases and influences decisions in scientific and engineering contexts where control of pressure is critical.

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Most popular questions from this chapter

The quantum states available for gas atoms of energy \(\epsilon\) in a cubical box of length \(L\) correspond to integer values for cach \(n_{x}, n_{y}\), and \(n_{z}\), according to Eq. (12.1). In a three-dimensional Euclidean space with coordinates \(n_{x}, n_{y}\), and \(n_{z+}\) each unit volume will contain one quantum state. The total number of quantum states \(\boldsymbol{g}^{\prime}\) with energy less than \(\epsilon^{\prime}\) is equal to the volume of the positive octant of a sphere of radius \(r=L\left(8 m \epsilon_{i}\right)^{1 / 2} / h\) (a) Show that $$ g^{\prime}=\frac{4 \pi V\left(2 m \epsilon^{\prime}\right)^{3 / 2}}{3 h^{3}} $$ (b) In a volume of \(1 \mathrm{~cm}^{3}\) of helium gas at \(300 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure, \(\epsilon^{\prime}\) is about \(10^{-5}\) J. Calculate \(g^{\prime}\). (c) Calculate the number \(N\) of helium atoms. (d) Show that \(g^{\prime} \gg N\).

The Doppler broadening of a spectral line increases with the rms speed of the atoms in the source of light. What should give narrower spectral lines: a mercury-198 lamp at \(300 \mathrm{~K}\) or a krypton-86 lamp at \(77 \mathrm{~K}\) ?

Given a gaseous system of \(N_{\mathrm{A}}\) indistinguishable, weakly interacting diatomic molecules: (a) Each molecule may vibrate with the same frequency \(v\), but with an energy \(\epsilon_{i}\), given by $$ \epsilon_{i}=\left(\frac{1}{2}+i\right) h v \quad(i=0,1,2, \cdots) $$ Show that the vibrational partition function \(Z_{y}\) is $$ Z_{v}=\frac{e^{-h v / 2 k T}}{1-e^{-h v / k T}} $$ (b) Each molecule may rotate, and the rotational partition function \(Z\), has the same form as that for translation, except that the volume \(V\) is replaced by the total solid angle \(4 \pi\), the mass is replaced by the moment of inertia 1, and the exponent \(\frac{3}{2}\) (referring to three translational degrees of freedom) is replaced by \(\frac{2}{2}\) since there are only two rotational degrees of freedom. Write the rotational partition function. (c) Taking into account translation, vibration, and rotation, calculate the Helmholtz function. (d) Calculate the pressure. (e) Calculate the internal energy. (f) Calculate the molar heat capacity at constant volume.

At what temperature is the mean translational kinetic energy of an atom equal to that of a singly charged ion of the same mass which has been accelerated from rest through a potential difference of: \((a) 1 \mathrm{~V} ?(b) 1,000 \mathrm{~V} ?(c) 1,000,000 \mathrm{~V} ?\) (Note: Neglect relativistic effects.)

Show that, when \(N\) ideal-gas atoms come to equilibrium, and $$ \begin{aligned} &\frac{g_{i}}{N_{i}}=\frac{Z}{N} e^{c_{1} / k T} \\ &\frac{Z}{N}=\frac{(k T)^{5 / 2}}{P}\left(\frac{2 \pi m}{h^{2}}\right)^{3 / 2} \end{aligned} $$ Taking \(\epsilon_{i}=\frac{3}{2} k T, T=300 K, P=10^{3} \mathrm{~Pa}\), and \(m=10^{-26} \mathrm{~kg}_{1}\) calculate \(\boldsymbol{g}_{i} / N_{i} .\)
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